java.lang.Object
- org.apfloat.ApcomplexMath

```
public class ApcomplexMath
extends java.lang.Object
```
Various mathematical functions for arbitrary precision complex numbers.

Version:

1.14.0

See Also:

ApfloatMath

Constructor Summary

Constructors
Modifier Constructor Description

private ApcomplexMath()

Method Summary

All Methods Static Methods Concrete Methods Deprecated Methods
Modifier and Type	Method	Description
`static Apfloat`	`abs(Apcomplex z)`	Absolute value.
`static Apcomplex`	`acos(Apcomplex z)`	Inverse cosine.
`(package private) static Apcomplex`	`acos(Apcomplex z, long precision)`
`static Apcomplex`	`acosh(Apcomplex z)`	Inverse hyperbolic cosine.
`(package private) static Apcomplex`	`acosh(Apcomplex z, long precision)`
`static Apcomplex`	`agm(Apcomplex a, Apcomplex b)`	Arithmetic-geometric mean.
`(package private) static Apcomplex`	`agm(Apcomplex a, Apcomplex b, java.util.function.Consumer<Apcomplex> consumer)`
`private static Apcomplex`	`agmConsume(java.util.function.Consumer<Apcomplex> consumer, Apcomplex a, Apcomplex c2, long workingPrecision)`
`private static Apcomplex`	`airy(java.util.function.Function<java.lang.Long,Apcomplex> f, long targetPrecision, int radix)`
`static Apcomplex`	`airyAi(Apcomplex z)`	Airy function Ai.
`(package private) static Apcomplex`	`airyAi(Apcomplex z0, long targetPrecision)`
`static Apcomplex`	`airyAiPrime(Apcomplex z)`	Derivative of the Airy function Ai.
`(package private) static Apcomplex`	`airyAiPrime(Apcomplex z0, long targetPrecision)`
`static Apcomplex`	`airyBi(Apcomplex z)`	Airy function Bi.
`(package private) static Apcomplex`	`airyBi(Apcomplex z0, long targetPrecision)`
`static Apcomplex`	`airyBiPrime(Apcomplex z)`	Derivative of the Airy function Bi.
`(package private) static Apcomplex`	`airyBiPrime(Apcomplex z0, long targetPrecision)`
`static Apcomplex[]`	`allRoots(Apcomplex z, int n)`	All values of the positive integer root.
`static Apfloat`	`arg(Apcomplex z)`	Angle of the complex vector in the complex plane.
`static Apcomplex`	`asin(Apcomplex z)`	Inverse sine.
`static Apcomplex`	`asinh(Apcomplex z)`	Inverse hyperbolic sine.
`static Apcomplex`	`atan(Apcomplex z)`	Inverse tangent.
`static Apcomplex`	`atanh(Apcomplex z)`	Inverse hyperbolic tangent.
`static Apcomplex`	`bernoulliB(long n, Apcomplex z)`	Bernoulli polynomial.
`(package private) static Apcomplex`	`bernoulliB(long n, Apcomplex z, long precision)`
`static Apcomplex`	`besselI(Apcomplex ν, Apcomplex z)`	Modified Bessel function of the first kind.
`static Apcomplex`	`besselJ(Apcomplex ν, Apcomplex z)`	Bessel function of the first kind.
`static Apcomplex`	`besselK(Apcomplex ν, Apcomplex z)`	Modified Bessel function of the second kind.
`static Apcomplex`	`besselY(Apcomplex ν, Apcomplex z)`	Bessel function of the second kind.
`static Apcomplex`	`beta(Apcomplex a, Apcomplex b)`	Beta function.
`static Apcomplex`	`beta(Apcomplex z, Apcomplex a, Apcomplex b)`	Incomplete beta function.
`static Apcomplex`	`beta(Apcomplex z1, Apcomplex z2, Apcomplex a, Apcomplex b)`	Generalized incomplete beta function.
`static Apcomplex`	`binomial(Apcomplex n, Apcomplex k)`	Binomial coefficient.
`static Apcomplex`	`cbrt(Apcomplex z)`	Cube root.
`static Apcomplex`	`chebyshevT(Apcomplex ν, Apcomplex z)`	Chebyshev function of the first kind.
`static Apcomplex`	`chebyshevU(Apcomplex ν, Apcomplex z)`	Chebyshev function of the second kind.
`static Apcomplex`	`cos(Apcomplex z)`	Cosine.
`static Apcomplex`	`cosh(Apcomplex z)`	Hyperbolic cosine.
`static Apcomplex`	`coshIntegral(Apcomplex z)`	Hyperbolic cosine integral.
`static Apcomplex`	`cosIntegral(Apcomplex z)`	Cosine integral.
`(package private) static Apcomplex`	`cot(Apcomplex z)`
`static Apcomplex`	`digamma(Apcomplex z)`	Digamma function.
`static Apcomplex`	`ellipticE(Apcomplex z)`	Complete elliptic integral of the second kind.
`(package private) static Apcomplex`	`ellipticE(Apcomplex z, long precision)`
`static Apcomplex`	`ellipticK(Apcomplex z)`	Complete elliptic integral of the first kind.
`(package private) static Apcomplex`	`ellipticK(Apcomplex z, long precision)`
`(package private) static Apcomplex`	`ellipticK(Apcomplex z, long precision, java.util.function.Consumer<Apcomplex> consumer)`
`static Apcomplex`	`erf(Apcomplex z)`	Error function.
`static Apcomplex`	`erfc(Apcomplex z)`	Complementary error function.
`(package private) static Apcomplex`	`erfcFixedPrecision(Apcomplex z)`
`(package private) static Apcomplex`	`erfFixedPrecision(Apcomplex z)`
`static Apcomplex`	`erfi(Apcomplex z)`	Imaginary error function.
`(package private) static Apcomplex`	`erfiFixedPrecision(Apcomplex z)`
`static Apcomplex`	`eulerE(long n, Apcomplex z)`	Euler polynomial.
`(package private) static Apcomplex`	`eulerE(long n, Apcomplex z, long precision)`
`static Apcomplex`	`exp(Apcomplex z)`	Exponent function.
`static Apcomplex`	`expIntegralE(Apcomplex ν, Apcomplex z)`	Exponential integral E.
`static Apcomplex`	`expIntegralEi(Apcomplex z)`	Exponential integral Ei.
`private static Apcomplex`	`expNoLoP(Apcomplex z)`
`static Apcomplex`	`fibonacci(Apcomplex ν, Apcomplex z)`	Fibonacci function.
`static Apcomplex`	`fresnelC(Apcomplex z)`	Fresnel integral C.
`static Apcomplex`	`fresnelS(Apcomplex z)`	Fresnel integral S.
`private static Apcomplex`	`fresnelTerm(Apint one, Apfloat half, Apfloat invSqrtPi, Apcomplex iz2, Apcomplex iHalfPiZ2)`
`static Apcomplex`	`gamma(Apcomplex z)`	Gamma function.
`static Apcomplex`	`gamma(Apcomplex a, Apcomplex z)`	Incomplete gamma function.
`static Apcomplex`	`gamma(Apcomplex a, Apcomplex z0, Apcomplex z1)`	Generalized incomplete gamma function.
`private static Apcomplex`	`gegenbauerC(long n, Apcomplex λ, Apcomplex z)`
`static Apcomplex`	`gegenbauerC(Apcomplex ν, Apcomplex z)`	Renormalized Gegenbauer function.
`static Apcomplex`	`gegenbauerC(Apcomplex ν, Apcomplex λ, Apcomplex z)`	Gegenbauer function.
`static Apcomplex`	`harmonicNumber(Apcomplex z)`	Harmonic number.
`static Apcomplex`	`harmonicNumber(Apcomplex z, Apcomplex r)`	Generalized harmonic number.
`static Apcomplex`	`hermiteH(Apcomplex ν, Apcomplex z)`	Hermite function.
`static Apcomplex`	`hypergeometric0F1(Apcomplex a, Apcomplex z)`	Confluent hypergeometric function ₀F₁.
`static Apcomplex`	`hypergeometric0F1Regularized(Apcomplex a, Apcomplex z)`	Regularized confluent hypergeometric function ₀F̃₁.
`static Apcomplex`	`hypergeometric1F1(Apcomplex a, Apcomplex b, Apcomplex z)`	Kummer confluent hypergeometric function ₁F₁.
`static Apcomplex`	`hypergeometric1F1Regularized(Apcomplex a, Apcomplex b, Apcomplex z)`	Regularized Kummer confluent hypergeometric function ₁F̃₁.
`static Apcomplex`	`hypergeometric2F1(Apcomplex a, Apcomplex b, Apcomplex c, Apcomplex z)`	Hypergeometric function ₂F₁.
`static Apcomplex`	`hypergeometric2F1Regularized(Apcomplex a, Apcomplex b, Apcomplex c, Apcomplex z)`	Regularized hypergeometric function ₂F̃₁.
`static Apcomplex`	`hypergeometricU(Apcomplex a, Apcomplex b, Apcomplex z)`	Tricomi's confluent hypergeometric function U.
`static Apcomplex`	`inverseRoot(Apcomplex z, long n)`	Inverse positive integer root.
`static Apcomplex`	`inverseRoot(Apcomplex z, long n, long k)`	Inverse positive integer root.
`private static Apcomplex`	`inverseRootAbs(Apcomplex z, long n, long k)`
`(package private) static boolean`	`isNonPositiveInteger(Apcomplex z)`
`private static Apcomplex`	`jacobiP(long n, Apcomplex a, Apcomplex b, Apcomplex z)`
`static Apcomplex`	`jacobiP(Apcomplex ν, Apcomplex a, Apcomplex b, Apcomplex z)`	Jacobi function.
`static Apcomplex`	`laguerreL(Apcomplex ν, Apcomplex z)`	Laguerre function.
`static Apcomplex`	`laguerreL(Apcomplex ν, Apcomplex λ, Apcomplex z)`	Generalized Laguerre function.
`private static Apcomplex`	`lastIterationExtendPrecision(int iterations, int precisingIteration, Apcomplex z)`
`static Apcomplex`	`legendreP(Apcomplex ν, Apcomplex z)`	Legendre function.
`static Apcomplex`	`legendreP(Apcomplex ν, Apcomplex μ, Apcomplex z)`	Associated Legendre function of the first kind.
`static Apcomplex`	`legendreQ(Apcomplex ν, Apcomplex z)`	Legendre function of the second kind.
`static Apcomplex`	`legendreQ(Apcomplex ν, Apcomplex μ, Apcomplex z)`	Associated Legendre function of the second kind.
`private static Apcomplex`	`limitPrecision(Apcomplex z)`
`static Apcomplex`	`log(Apcomplex z)`	Natural logarithm.
`static Apcomplex`	`log(Apcomplex z, Apcomplex w)`	Logarithm in arbitrary base.
`static Apcomplex`	`logGamma(Apcomplex z)`	Logarithm of the gamma function.
`static Apcomplex`	`logIntegral(Apcomplex z)`	Logarithmic integral.
`static Apcomplex`	`logisticSigmoid(Apcomplex z)`	Logistic sigmoid.
`private static Apcomplex`	`logPochhammer(Apcomplex z, long n)`
`private static Apcomplex`	`logSin(Apcomplex z)`
`static Apcomplex`	`negate(Apcomplex z)`	Deprecated. Use `Apcomplex.negate()`.
`static Apfloat`	`norm(Apcomplex z)`	Norm.
`(package private) static Apcomplex`	`pochhammer(Apcomplex z, long n)`
`static Apcomplex`	`pochhammer(Apcomplex z, Apcomplex n)`	Pochhammer symbol.
`static Apcomplex`	`polygamma(long n, Apcomplex z)`	Polygamma function.
`static Apcomplex`	`polylog(Apcomplex ν, Apcomplex z)`	Polylogarithm.
`static Apcomplex`	`pow(Apcomplex z, long n)`	Integer power.
`static Apcomplex`	`pow(Apcomplex z, Apcomplex w)`	Arbitrary power.
`private static Apcomplex`	`powAbs(Apcomplex z, long n)`
`static Apcomplex`	`product(Apcomplex... z)`	Product of numbers.
`private static Apcomplex`	`rawLog(Apcomplex z)`
`private static Apcomplex`	`rightSqrt(Apcomplex z, Apcomplex reference)`
`static Apcomplex`	`root(Apcomplex z, long n)`	Positive integer root.
`static Apcomplex`	`root(Apcomplex z, long n, long k)`	Positive integer root.
`static Apcomplex`	`scale(Apcomplex z, long scale)`	Multiply by a power of the radix.
`static Apcomplex`	`sin(Apcomplex z)`	Sine.
`static Apcomplex`	`sinc(Apcomplex z)`	Sinc.
`static Apcomplex`	`sinh(Apcomplex z)`	Hyperbolic sine.
`static Apcomplex`	`sinhIntegral(Apcomplex z)`	Hyperbolic sine integral.
`static Apcomplex`	`sinIntegral(Apcomplex z)`	Sine integral.
`static Apcomplex`	`sphericalHarmonicY(Apcomplex λ, Apcomplex μ, Apcomplex ϑ, Apcomplex ϕ)`	Spherical harmonic function.
`private static Apcomplex`	`sphericalHarmonicY(Apint n, Apint m, Apcomplex ϑ, Apcomplex ϕ)`
`static Apcomplex`	`sqrt(Apcomplex z)`	Square root.
`static Apcomplex`	`sum(Apcomplex... z)`	Sum of numbers.
`static Apcomplex`	`tan(Apcomplex z)`	Tangent.
`(package private) static Apcomplex`	`tan(Apcomplex z, boolean negate)`
`(package private) static Apcomplex`	`tanFixedPrecision(Apcomplex z)`
`static Apcomplex`	`tanh(Apcomplex z)`	Hyperbolic tangent.
`private static Apcomplex`	`tanh(Apcomplex z, boolean negate)`
`(package private) static Apcomplex`	`tanhFixedPrecision(Apcomplex z)`
`static Apfloat`	`ulp(Apcomplex z)`	Returns the unit in the last place of the argument, considering the scale and precision.
`static Apcomplex`	`w(Apcomplex z)`	Lambert W function.
`static Apcomplex`	`w(Apcomplex z, long k)`	Lambert W function for the specified branch.
`static Apcomplex`	`zeta(Apcomplex s)`	Riemann zeta function.
`static Apcomplex`	`zeta(Apcomplex s, Apcomplex a)`	Hurwitz zeta function.

Methods inherited from class java.lang.Object
clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Detail
- ApcomplexMath
```
private ApcomplexMath()
```

Method Detail

negate

@Deprecated
public static Apcomplex negate(Apcomplex z)
                        throws ApfloatRuntimeException

Deprecated.

Use Apcomplex.negate().

Negative value.

Parameters:: z - The argument.
Returns:: -z.
Throws:: ApfloatRuntimeException

abs
```
public static Apfloat abs(Apcomplex z)
                   throws ApfloatRuntimeException
```
Absolute value.

Parameters:

z - The argument.

Returns:

sqrt(x² + y²), where z = x + i y.

Throws:

ApfloatRuntimeException

norm
```
public static Apfloat norm(Apcomplex z)
                    throws ApfloatRuntimeException
```
Norm. Square of the magnitude.

Parameters:

z - The argument.

Returns:

x² + y², where z = x + i y.

Throws:

ApfloatRuntimeException

arg
```
public static Apfloat arg(Apcomplex z)
                   throws java.lang.ArithmeticException,
                          ApfloatRuntimeException
```
Angle of the complex vector in the complex plane.

Parameters:

z - The argument.

Returns:

arctan(y / x) from the appropriate branch, where z = x + i y.

Throws:

java.lang.ArithmeticException - If z is zero.

ApfloatRuntimeException

scale

public static Apcomplex scale(Apcomplex z,
                              long scale)
                       throws ApfloatRuntimeException

Multiply by a power of the radix.

Parameters:: z - The argument.; scale - The scaling factor.
Returns:: z * z.radix()^scale.
Throws:: ApfloatRuntimeException

pow

public static Apcomplex pow(Apcomplex z,
                            long n)
                     throws java.lang.ArithmeticException,
                            ApfloatRuntimeException

Integer power.

Parameters:: z - Base of the power operator.; n - Exponent of the power operator.
Returns:: z to the n:th power, that is zⁿ.
Throws:: java.lang.ArithmeticException - If both z and n are zero.; ApfloatRuntimeException

powAbs

private static Apcomplex powAbs(Apcomplex z,
                                long n)
                         throws java.lang.ArithmeticException,
                                ApfloatRuntimeException

Throws:: java.lang.ArithmeticException; ApfloatRuntimeException

sqrt

public static Apcomplex sqrt(Apcomplex z)
                      throws ApfloatRuntimeException

Square root.

Parameters:: z - The argument.
Returns:: Square root of z.
Throws:: ApfloatRuntimeException

cbrt

public static Apcomplex cbrt(Apcomplex z)
                      throws ApfloatRuntimeException

Cube root.

Parameters:: z - The argument.
Returns:: Cube root of z.
Throws:: ApfloatRuntimeException

root
```
public static Apcomplex root(Apcomplex z,
                             long n)
                      throws java.lang.ArithmeticException,
                             ApfloatRuntimeException
```
Positive integer root. The branch that has the smallest angle and same sign of imaginary part as z is always chosen.

Parameters:

z - The argument.

n - Which root to take.

Returns:

n:th root of z, that is z^1/n.

Throws:

java.lang.ArithmeticException - If n is zero.

ApfloatRuntimeException

root
```
public static Apcomplex root(Apcomplex z,
                             long n,
                             long k)
                      throws java.lang.ArithmeticException,
                             ApfloatRuntimeException
```
Positive integer root. The specified branch counting from the smallest angle and same sign of imaginary part as z is chosen.

Parameters:

z - The argument.

n - Which root to take.

k - Which branch to take.

Returns:

n:th root of z, that is z^1/ne^i2πsk/n where s is the signum of the imaginary part of z.

Throws:

java.lang.ArithmeticException - If n is zero.

ApfloatRuntimeException

Since:

1.5

inverseRoot
```
public static Apcomplex inverseRoot(Apcomplex z,
                                    long n)
                             throws java.lang.ArithmeticException,
                                    ApfloatRuntimeException
```
Inverse positive integer root. The branch that has the smallest angle and different sign of imaginary part than z is always chosen.

Parameters:

z - The argument.

n - Which inverse root to take.

Returns:

Inverse n:th root of z, that is z^-1/n.

Throws:

java.lang.ArithmeticException - If z or n is zero.

ApfloatRuntimeException

inverseRoot
```
public static Apcomplex inverseRoot(Apcomplex z,
                                    long n,
                                    long k)
                             throws java.lang.ArithmeticException,
                                    ApfloatRuntimeException
```
Inverse positive integer root. The specified branch counting from the smallest angle and different sign of imaginary part than z is chosen.

Parameters:

z - The argument.

n - Which inverse root to take.

k - Which branch to take.

Returns:

Inverse n:th root of z, that is z^-1/ne^-i2πk/n.

Throws:

java.lang.ArithmeticException - If z or n is zero.

ApfloatRuntimeException

inverseRootAbs

private static Apcomplex inverseRootAbs(Apcomplex z,
                                        long n,
                                        long k)
                                 throws java.lang.ArithmeticException,
                                        ApfloatRuntimeException

Throws:: java.lang.ArithmeticException; ApfloatRuntimeException

allRoots
```
public static Apcomplex[] allRoots(Apcomplex z,
                                   int n)
                            throws java.lang.ArithmeticException,
                                   ApfloatRuntimeException
```
All values of the positive integer root.
Returns all of the n values of the root, in the order of the angle, starting from the smallest angle and same sign of imaginary part as z.

Parameters:

z - The argument.

n - Which root to take.

Returns:

All values of the n:th root of z, that is z^1/n, in the order of the angle.

Throws:

java.lang.ArithmeticException - If n is zero.

ApfloatRuntimeException

Since:

1.5

agm

public static Apcomplex agm(Apcomplex a,
                            Apcomplex b)
                     throws ApfloatRuntimeException

Arithmetic-geometric mean.

Parameters:: a - First argument.; b - Second argument.
Returns:: Arithmetic-geometric mean of a and b.
Throws:: ApfloatRuntimeException

agm

static Apcomplex agm(Apcomplex a,
                     Apcomplex b,
                     java.util.function.Consumer<Apcomplex> consumer)
              throws ApfloatRuntimeException

Throws:: ApfloatRuntimeException

limitPrecision

private static Apcomplex limitPrecision(Apcomplex z)

rightSqrt

private static Apcomplex rightSqrt(Apcomplex z,
                                   Apcomplex reference)

agmConsume

private static Apcomplex agmConsume(java.util.function.Consumer<Apcomplex> consumer,
                                    Apcomplex a,
                                    Apcomplex c2,
                                    long workingPrecision)

log
```
public static Apcomplex log(Apcomplex z)
                     throws java.lang.ArithmeticException,
                            ApfloatRuntimeException
```
Natural logarithm.
The logarithm is calculated using the arithmetic-geometric mean. See the Borweins' book for the formula.

Parameters:

z - The argument.

Returns:

Natural logarithm of z.

Throws:

java.lang.ArithmeticException - If z is zero.

ApfloatRuntimeException

log

public static Apcomplex log(Apcomplex z,
                            Apcomplex w)
                     throws java.lang.ArithmeticException,
                            ApfloatRuntimeException

Logarithm in arbitrary base.

Parameters:: z - The argument.; w - The base.
Returns:: Base-w logarithm of z.
Throws:: java.lang.ArithmeticException - If z or w is zero.; ApfloatRuntimeException
Since:: 1.6

rawLog

private static Apcomplex rawLog(Apcomplex z)
                         throws ApfloatRuntimeException

Throws:: ApfloatRuntimeException

exp
```
public static Apcomplex exp(Apcomplex z)
                     throws ApfloatRuntimeException
```
Exponent function. Calculated using Newton's iteration for the inverse of logarithm.

Parameters:

z - The argument.

Returns:

e^z.

Throws:

ApfloatRuntimeException

pow
```
public static Apcomplex pow(Apcomplex z,
                            Apcomplex w)
                     throws ApfloatRuntimeException
```
Arbitrary power. Calculated using log() and exp().

Parameters:

z - The base.

w - The exponent.

Returns:

z^w.

Throws:

java.lang.ArithmeticException - If both z and w are zero.

ApfloatRuntimeException

acos
```
public static Apcomplex acos(Apcomplex z)
                      throws ApfloatRuntimeException
```
Inverse cosine. Calculated using log().

Parameters:

z - The argument.

Returns:

Inverse cosine of z.

Throws:

ApfloatRuntimeException

acos

static Apcomplex acos(Apcomplex z,
                      long precision)

acosh
```
public static Apcomplex acosh(Apcomplex z)
                       throws ApfloatRuntimeException
```
Inverse hyperbolic cosine. Calculated using log().

Parameters:

z - The argument.

Returns:

Inverse hyperbolic cosine of z.

Throws:

ApfloatRuntimeException

acosh

static Apcomplex acosh(Apcomplex z,
                       long precision)

asin
```
public static Apcomplex asin(Apcomplex z)
                      throws ApfloatRuntimeException
```
Inverse sine. Calculated using log().

Parameters:

z - The argument.

Returns:

Inverse sine of z.

Throws:

ApfloatRuntimeException

asinh
```
public static Apcomplex asinh(Apcomplex z)
                       throws ApfloatRuntimeException
```
Inverse hyperbolic sine. Calculated using log().

Parameters:

z - The argument.

Returns:

Inverse hyperbolic sine of z.

Throws:

ApfloatRuntimeException

atan

public static Apcomplex atan(Apcomplex z)
                      throws java.lang.ArithmeticException,
                             ApfloatRuntimeException

Inverse tangent. Calculated using log().

Parameters:: z - The argument.
Returns:: Inverse tangent of z.
Throws:: java.lang.ArithmeticException - If z == i.; ApfloatRuntimeException

atanh

public static Apcomplex atanh(Apcomplex z)
                       throws java.lang.ArithmeticException,
                              ApfloatRuntimeException

Inverse hyperbolic tangent. Calculated using log().

Parameters:: z - The argument.
Returns:: Inverse hyperbolic tangent of z.
Throws:: java.lang.ArithmeticException - If z is 1 or -1.; ApfloatRuntimeException

cos
```
public static Apcomplex cos(Apcomplex z)
                     throws ApfloatRuntimeException
```
Cosine. Calculated using exp().

Parameters:

z - The argument.

Returns:

Cosine of z.

Throws:

ApfloatRuntimeException

cosh
```
public static Apcomplex cosh(Apcomplex z)
                      throws ApfloatRuntimeException
```
Hyperbolic cosine. Calculated using exp().

Parameters:

z - The argument.

Returns:

Hyperbolic cosine of z.

Throws:

ApfloatRuntimeException

sin

public static Apcomplex sin(Apcomplex z)
                     throws ApfloatRuntimeException

Sine. Calculated using exp().

Parameters:: z - The argument.
Returns:: Sine of z.
Throws:: ApfloatRuntimeException

sinh
```
public static Apcomplex sinh(Apcomplex z)
                      throws ApfloatRuntimeException
```
Hyperbolic sine. Calculated using exp().

Parameters:

z - The argument.

Returns:

Hyperbolic sine of z.

Throws:

ApfloatRuntimeException

tan

public static Apcomplex tan(Apcomplex z)
                     throws java.lang.ArithmeticException,
                            ApfloatRuntimeException

Tangent. Calculated using exp().

Parameters:: z - The argument.
Returns:: Tangent of z.
Throws:: java.lang.ArithmeticException - If z is π/2 + n π where n is an integer.; ApfloatRuntimeException

tanFixedPrecision

static Apcomplex tanFixedPrecision(Apcomplex z)
                            throws java.lang.ArithmeticException,
                                   ApfloatRuntimeException

Throws:: java.lang.ArithmeticException; ApfloatRuntimeException

tan

static Apcomplex tan(Apcomplex z,
                     boolean negate)
              throws java.lang.ArithmeticException,
                     ApfloatRuntimeException

Throws:: java.lang.ArithmeticException; ApfloatRuntimeException

tanh
```
public static Apcomplex tanh(Apcomplex z)
                      throws java.lang.ArithmeticException,
                             ApfloatRuntimeException
```
Hyperbolic tangent. Calculated using exp().

Parameters:

z - The argument.

Returns:

Hyperbolic tangent of z.

Throws:

java.lang.ArithmeticException - If z is i (π/2 + n π) where n is an integer.

ApfloatRuntimeException

tanhFixedPrecision

static Apcomplex tanhFixedPrecision(Apcomplex z)
                             throws java.lang.ArithmeticException,
                                    ApfloatRuntimeException

Throws:: java.lang.ArithmeticException; ApfloatRuntimeException

tanh

private static Apcomplex tanh(Apcomplex z,
                              boolean negate)
                       throws java.lang.ArithmeticException,
                              ApfloatRuntimeException

Throws:: java.lang.ArithmeticException; ApfloatRuntimeException

cot

static Apcomplex cot(Apcomplex z)
              throws java.lang.ArithmeticException,
                     ApfloatRuntimeException

Throws:: java.lang.ArithmeticException; ApfloatRuntimeException

sinc

public static Apcomplex sinc(Apcomplex z)
                      throws ApfloatRuntimeException

Sinc.

Parameters:: z - The argument.
Returns:: sinc(z)
Throws:: ApfloatRuntimeException
Since:: 1.14.0

w
```
public static Apcomplex w(Apcomplex z)
                   throws ApfloatRuntimeException
```
Lambert W function. The W function gives the solution to the equation W e^W = z. Also known as the product logarithm.
This function gives the solution to the principal branch, W₀.

Parameters:

z - The argument.

Returns:

W₀(z).

Throws:

ApfloatRuntimeException

Since:

1.8.0

w
```
public static Apcomplex w(Apcomplex z,
                          long k)
                   throws java.lang.ArithmeticException,
                          ApfloatRuntimeException
```
Lambert W function for the specified branch.

Parameters:

z - The argument.

k - The branch.

Returns:

W_k(z).

Throws:

java.lang.ArithmeticException - If z is zero and k is not zero.

ApfloatRuntimeException

Since:

1.8.0

See Also:

w(Apcomplex)

product
```
public static Apcomplex product(Apcomplex... z)
                         throws ApfloatRuntimeException
```
Product of numbers. The precision used in the multiplications is only what is needed for the end result. This method may perform significantly better than simply multiplying the numbers sequentially.
If there are no arguments, the return value is 1.

Parameters:

z - The argument(s).

Returns:

The product of the given numbers.

Throws:

ApfloatRuntimeException

Since:

1.3

sum
```
public static Apcomplex sum(Apcomplex... z)
                     throws ApfloatRuntimeException
```
Sum of numbers. The precision used in the additions is only what is needed for the end result. This method may perform significantly better than simply adding the numbers sequentially.
If there are no arguments, the return value is 0.

Parameters:

z - The argument(s).

Returns:

The sum of the given numbers.

Throws:

ApfloatRuntimeException

Since:

1.3

gamma

public static Apcomplex gamma(Apcomplex z)
                       throws java.lang.ArithmeticException,
                              ApfloatRuntimeException

Gamma function.

Parameters:: z - The argument.
Returns:: Γ(z)
Throws:: java.lang.ArithmeticException - If z is a nonpositive integer.; ApfloatRuntimeException
Since:: 1.9.0

gamma

public static Apcomplex gamma(Apcomplex a,
                              Apcomplex z)
                       throws java.lang.ArithmeticException,
                              ApfloatRuntimeException

Incomplete gamma function.

Parameters:: a - The first argument.; z - The second argument.
Returns:: Γ(a, z)
Throws:: java.lang.ArithmeticException - If the real part of a is nonpositive and z is zero.; ApfloatRuntimeException
Since:: 1.10.0

gamma
```
public static Apcomplex gamma(Apcomplex a,
                              Apcomplex z0,
                              Apcomplex z1)
                       throws java.lang.ArithmeticException,
                              ApfloatRuntimeException
```
Generalized incomplete gamma function.
This function is defined as: Γ(a, z0, z1) = Γ(a, z0) - Γ(a, z1)
The lower gamma function can be calculated with: γ(a, z) = Γ(a, 0, z)

Parameters:

a - The first argument.

z0 - The second argument.

z1 - The third argument.

Returns:

Γ(a, z0, z1)

Throws:

java.lang.ArithmeticException - If the real part of a is nonpositive and either z0 or z1 is zero. For the lower gamma function if a is a nonpositive integer.

ApfloatRuntimeException

Since:

1.10.0

logGamma
```
public static Apcomplex logGamma(Apcomplex z)
                          throws java.lang.ArithmeticException,
                                 ApfloatRuntimeException
```
Logarithm of the gamma function. Note that this function has a different branch structure than log(gamma(z)).

Parameters:

z - The argument.

Returns:

logΓ(z)

Throws:

java.lang.ArithmeticException - If z is a nonpositive integer.

ApfloatRuntimeException

Since:

1.11.0

logSin

private static Apcomplex logSin(Apcomplex z)

expNoLoP

private static Apcomplex expNoLoP(Apcomplex z)

logPochhammer

private static Apcomplex logPochhammer(Apcomplex z,
                                       long n)

digamma

public static Apcomplex digamma(Apcomplex z)
                         throws java.lang.ArithmeticException,
                                ApfloatRuntimeException

Digamma function.

Parameters:: z - The argument.
Returns:: ψ(z)
Throws:: java.lang.ArithmeticException - If z is a nonpositive integer.; ApfloatRuntimeException
Since:: 1.11.0

polygamma

public static Apcomplex polygamma(long n,
                                  Apcomplex z)
                           throws java.lang.ArithmeticException,
                                  ApfloatRuntimeException

Polygamma function.

Parameters:: n - The order.; z - The argument.
Returns:: ψ⁽ⁿ⁾(z)
Throws:: java.lang.ArithmeticException - If n is negative or z is a nonpositive integer.; ApfloatRuntimeException
Since:: 1.13.0

beta

public static Apcomplex beta(Apcomplex a,
                             Apcomplex b)
                      throws java.lang.ArithmeticException,
                             ApfloatRuntimeException

Beta function.

Parameters:: a - The first argument.; b - The second argument.
Returns:: B(a, b)
Throws:: java.lang.ArithmeticException - If a or b is a nonpositive integer but a + b is not. Also if both a and b are nonpositive integers.; ApfloatRuntimeException
Since:: 1.13.0

beta

public static Apcomplex beta(Apcomplex z,
                             Apcomplex a,
                             Apcomplex b)
                      throws java.lang.ArithmeticException,
                             ApfloatRuntimeException

Incomplete beta function.

Parameters:: z - The first argument.; a - The second argument.; b - The third argument.
Returns:: B_z(a, b)
Throws:: java.lang.ArithmeticException - If a is a nonpositive integer or z is zero and a has nonpositive real part.; ApfloatRuntimeException
Since:: 1.13.0

beta

public static Apcomplex beta(Apcomplex z1,
                             Apcomplex z2,
                             Apcomplex a,
                             Apcomplex b)
                      throws java.lang.ArithmeticException,
                             ApfloatRuntimeException

Generalized incomplete beta function.

Parameters:: z1 - The first argument.; z2 - The second argument.; a - The third argument.; b - The fourth argument.
Returns:: B_{(z1, z2)}(a, b)
Throws:: java.lang.ArithmeticException - If a is a nonpositive integer or z1 or z2 is zero and a has nonpositive real part.; ApfloatRuntimeException
Since:: 1.13.0

pochhammer

public static Apcomplex pochhammer(Apcomplex z,
                                   Apcomplex n)
                            throws java.lang.ArithmeticException,
                                   ApfloatRuntimeException

Pochhammer symbol.

Parameters:: z - The first argument.; n - The second argument.
Returns:: (z)_n
Throws:: java.lang.ArithmeticException - If z + n is a nonpositive integer but z is not.; ApfloatRuntimeException
Since:: 1.13.0

pochhammer

static Apcomplex pochhammer(Apcomplex z,
                            long n)

binomial

public static Apcomplex binomial(Apcomplex n,
                                 Apcomplex k)
                          throws java.lang.ArithmeticException,
                                 ApfloatRuntimeException

Binomial coefficient. Calculated using the gamma(Apcomplex) function.

Parameters:: n - The first argument.; k - The second argument.
Returns:: $(\binom{n}{k})$
Throws:: java.lang.ArithmeticException - If n is a negative integer and k is noninteger.; ApfloatRuntimeException
Since:: 1.11.0

zeta

public static Apcomplex zeta(Apcomplex s)
                      throws java.lang.ArithmeticException,
                             ApfloatRuntimeException

Riemann zeta function.

Parameters:: s - The argument.
Returns:: ζ(s)
Throws:: java.lang.ArithmeticException - If s is 1.; ApfloatRuntimeException
Since:: 1.11.0

zeta

public static Apcomplex zeta(Apcomplex s,
                             Apcomplex a)
                      throws java.lang.ArithmeticException,
                             ApfloatRuntimeException

Hurwitz zeta function.

Parameters:: s - The first argument.; a - The second argument.
Returns:: ζ(s, a)
Throws:: java.lang.ArithmeticException - If s is 1 or if a is a nonpositive integer.; ApfloatRuntimeException
Since:: 1.11.0

hypergeometric0F1

public static Apcomplex hypergeometric0F1(Apcomplex a,
                                          Apcomplex z)
                                   throws java.lang.ArithmeticException,
                                          ApfloatRuntimeException

Confluent hypergeometric function ₀F₁.

Parameters:: a - The first argument.; z - The second argument.
Returns:: ₀F₁(; a; z)
Throws:: java.lang.ArithmeticException - If the function value is not finite.; ApfloatRuntimeException
Since:: 1.11.0

hypergeometric0F1Regularized

public static Apcomplex hypergeometric0F1Regularized(Apcomplex a,
                                                     Apcomplex z)
                                              throws ApfloatRuntimeException

Regularized confluent hypergeometric function ₀F̃₁.

Parameters:: a - The first argument.; z - The second argument.
Returns:: ₀F̃₁(; a; z)
Throws:: ApfloatRuntimeException
Since:: 1.13.0

hypergeometric1F1

public static Apcomplex hypergeometric1F1(Apcomplex a,
                                          Apcomplex b,
                                          Apcomplex z)
                                   throws java.lang.ArithmeticException,
                                          ApfloatRuntimeException

Kummer confluent hypergeometric function ₁F₁. Also known as the confluent hypergeometric function of the first kind.

Parameters:: a - The first argument.; b - The second argument.; z - The third argument.
Returns:: ₁F₁(a; b; z)
Throws:: java.lang.ArithmeticException - If the function value is not finite.; ApfloatRuntimeException
Since:: 1.11.0

hypergeometric1F1Regularized

public static Apcomplex hypergeometric1F1Regularized(Apcomplex a,
                                                     Apcomplex b,
                                                     Apcomplex z)
                                              throws ApfloatRuntimeException

Regularized Kummer confluent hypergeometric function ₁F̃₁. Also known as the regularized confluent hypergeometric function of the first kind.

Parameters:: a - The first argument.; b - The second argument.; z - The third argument.
Returns:: ₁F̃₁(a; b; z)
Throws:: ApfloatRuntimeException
Since:: 1.13.0

hypergeometric2F1

public static Apcomplex hypergeometric2F1(Apcomplex a,
                                          Apcomplex b,
                                          Apcomplex c,
                                          Apcomplex z)
                                   throws java.lang.ArithmeticException,
                                          ApfloatRuntimeException

Hypergeometric function ₂F₁. Also known as the Gaussian or ordinary hypergeometric function.

Parameters:: a - The first argument.; b - The second argument.; c - The third argument.; z - The fourth argument.
Returns:: ₂F₁(a, b; c; z)
Throws:: java.lang.ArithmeticException - If the function value is not finite.; ApfloatRuntimeException
Since:: 1.11.0

hypergeometric2F1Regularized

public static Apcomplex hypergeometric2F1Regularized(Apcomplex a,
                                                     Apcomplex b,
                                                     Apcomplex c,
                                                     Apcomplex z)
                                              throws ApfloatRuntimeException

Regularized hypergeometric function ₂F̃₁. Also known as the regularized Gaussian or ordinary hypergeometric function.

Parameters:: a - The first argument.; b - The second argument.; c - The third argument.; z - The fourth argument.
Returns:: ₂F̃₁(a, b; c; z)
Throws:: ApfloatRuntimeException
Since:: 1.13.0

hypergeometricU

public static Apcomplex hypergeometricU(Apcomplex a,
                                        Apcomplex b,
                                        Apcomplex z)
                                 throws java.lang.ArithmeticException,
                                        ApfloatRuntimeException

Tricomi's confluent hypergeometric function U. Also known as the confluent hypergeometric function of the second kind.

Parameters:: a - The first argument.; b - The second argument.; z - The third argument.
Returns:: U(a, b, z)
Throws:: java.lang.ArithmeticException - If the result is not finite.; ApfloatRuntimeException
Since:: 1.13.0

erf

public static Apcomplex erf(Apcomplex z)
                     throws ApfloatRuntimeException

Error function.

Parameters:: z - The argument.
Returns:: erf(z)
Throws:: ApfloatRuntimeException
Since:: 1.13.0

erfFixedPrecision

static Apcomplex erfFixedPrecision(Apcomplex z)
                            throws ApfloatRuntimeException

Throws:: ApfloatRuntimeException

erfc
```
public static Apcomplex erfc(Apcomplex z)
                      throws ApfloatRuntimeException
```
Complementary error function.

Parameters:

z - The argument.

Returns:

erfc(z)

Throws:

ApfloatRuntimeException

Since:

1.13.0

erfcFixedPrecision

static Apcomplex erfcFixedPrecision(Apcomplex z)
                             throws ApfloatRuntimeException

Throws:: ApfloatRuntimeException

erfi
```
public static Apcomplex erfi(Apcomplex z)
                      throws ApfloatRuntimeException
```
Imaginary error function.

Parameters:

z - The argument.

Returns:

erfi(z)

Throws:

ApfloatRuntimeException

Since:

1.13.0

erfiFixedPrecision

static Apcomplex erfiFixedPrecision(Apcomplex z)
                             throws ApfloatRuntimeException

Throws:: ApfloatRuntimeException

fresnelS

public static Apcomplex fresnelS(Apcomplex z)
                          throws ApfloatRuntimeException

Fresnel integral S.

Parameters:: z - The argument.
Returns:: S(z)
Throws:: ApfloatRuntimeException
Since:: 1.13.0

fresnelC

public static Apcomplex fresnelC(Apcomplex z)
                          throws ApfloatRuntimeException

Fresnel integral C.

Parameters:: z - The argument.
Returns:: C(z)
Throws:: ApfloatRuntimeException
Since:: 1.13.0

fresnelTerm

private static Apcomplex fresnelTerm(Apint one,
                                     Apfloat half,
                                     Apfloat invSqrtPi,
                                     Apcomplex iz2,
                                     Apcomplex iHalfPiZ2)
                              throws ApfloatRuntimeException

Throws:: ApfloatRuntimeException

expIntegralE

public static Apcomplex expIntegralE(Apcomplex ν,
                                     Apcomplex z)
                              throws java.lang.ArithmeticException,
                                     ApfloatRuntimeException

Exponential integral E.

Parameters:: \u03bd - The first argument.; z - The second argument.
Returns:: E_ν(z)
Throws:: java.lang.ArithmeticException - If real part of ν is ≤ 1 and z is zero.; ApfloatRuntimeException
Since:: 1.13.0

expIntegralEi

public static Apcomplex expIntegralEi(Apcomplex z)
                               throws java.lang.ArithmeticException,
                                      ApfloatRuntimeException

Exponential integral Ei.

Parameters:: z - The argument.
Returns:: Ei(z)
Throws:: java.lang.ArithmeticException - If z is zero.; ApfloatRuntimeException
Since:: 1.13.0

logIntegral

public static Apcomplex logIntegral(Apcomplex z)
                             throws java.lang.ArithmeticException,
                                    ApfloatRuntimeException

Logarithmic integral.

Parameters:: z - The argument.
Returns:: li(z)
Throws:: java.lang.ArithmeticException - If z is zero.; ApfloatRuntimeException
Since:: 1.13.0

sinIntegral

public static Apcomplex sinIntegral(Apcomplex z)
                             throws ApfloatRuntimeException

Sine integral.

Parameters:: z - The argument.
Returns:: Si(z)
Throws:: ApfloatRuntimeException
Since:: 1.13.0

cosIntegral

public static Apcomplex cosIntegral(Apcomplex z)
                             throws java.lang.ArithmeticException,
                                    ApfloatRuntimeException

Cosine integral.

Parameters:: z - The argument.
Returns:: Ci(z)
Throws:: java.lang.ArithmeticException - If z is zero.; ApfloatRuntimeException
Since:: 1.13.0

sinhIntegral

public static Apcomplex sinhIntegral(Apcomplex z)
                              throws ApfloatRuntimeException

Hyperbolic sine integral.

Parameters:: z - The argument.
Returns:: Shi(z)
Throws:: ApfloatRuntimeException
Since:: 1.13.0

coshIntegral

public static Apcomplex coshIntegral(Apcomplex z)
                              throws java.lang.ArithmeticException,
                                     ApfloatRuntimeException

Hyperbolic cosine integral.

Parameters:: z - The argument.
Returns:: Chi(z)
Throws:: java.lang.ArithmeticException - If z is zero.; ApfloatRuntimeException
Since:: 1.13.0

airyAi
```
public static Apcomplex airyAi(Apcomplex z)
                        throws ApfloatRuntimeException
```
Airy function Ai.

Parameters:

z - The argument.

Returns:

Ai(z)

Throws:

InfiniteExpansionException - If z is zero.

ApfloatRuntimeException

Since:

1.13.0

airyAi

static Apcomplex airyAi(Apcomplex z0,
                        long targetPrecision)
                 throws ApfloatRuntimeException

Throws:: ApfloatRuntimeException

airyAiPrime
```
public static Apcomplex airyAiPrime(Apcomplex z)
                             throws ApfloatRuntimeException
```
Derivative of the Airy function Ai.

Parameters:

z - The argument.

Returns:

Ai′(z)

Throws:

InfiniteExpansionException - If z is zero.

ApfloatRuntimeException

Since:

1.13.0

airyAiPrime

static Apcomplex airyAiPrime(Apcomplex z0,
                             long targetPrecision)
                      throws ApfloatRuntimeException

Throws:: ApfloatRuntimeException

airyBi
```
public static Apcomplex airyBi(Apcomplex z)
                        throws ApfloatRuntimeException
```
Airy function Bi.

Parameters:

z - The argument.

Returns:

Bi(z)

Throws:

InfiniteExpansionException - If z is zero.

ApfloatRuntimeException

Since:

1.13.0

airyBi

static Apcomplex airyBi(Apcomplex z0,
                        long targetPrecision)
                 throws ApfloatRuntimeException

Throws:: ApfloatRuntimeException

airyBiPrime
```
public static Apcomplex airyBiPrime(Apcomplex z)
                             throws ApfloatRuntimeException
```
Derivative of the Airy function Bi.

Parameters:

z - The argument.

Returns:

Bi′(z)

Throws:

InfiniteExpansionException - If z is zero.

ApfloatRuntimeException

Since:

1.13.0

airyBiPrime

static Apcomplex airyBiPrime(Apcomplex z0,
                             long targetPrecision)
                      throws ApfloatRuntimeException

Throws:: ApfloatRuntimeException

airy

private static Apcomplex airy(java.util.function.Function<java.lang.Long,Apcomplex> f,
                              long targetPrecision,
                              int radix)

besselJ
```
public static Apcomplex besselJ(Apcomplex ν,
                                Apcomplex z)
                         throws java.lang.ArithmeticException,
                                ApfloatRuntimeException
```
Bessel function of the first kind.

Parameters:

\u03bd - The order.

z - The argument.

Returns:

J_ν(z)

Throws:

java.lang.ArithmeticException - If the real part of ν is < 0 and ν is not an integer and z is zero. Also if the real part of ν is zero but the imaginary part is not, and z is zero.

ApfloatRuntimeException

Since:

1.13.0

besselI
```
public static Apcomplex besselI(Apcomplex ν,
                                Apcomplex z)
                         throws java.lang.ArithmeticException,
                                ApfloatRuntimeException
```
Modified Bessel function of the first kind.

Parameters:

\u03bd - The order.

z - The argument.

Returns:

I_ν(z)

Throws:

java.lang.ArithmeticException - If the real part of ν is < 0 and ν is not an integer and z is zero. Also if the real part of ν is zero but the imaginary part is not, and z is zero.

ApfloatRuntimeException

Since:

1.13.0

besselY

public static Apcomplex besselY(Apcomplex ν,
                                Apcomplex z)
                         throws java.lang.ArithmeticException,
                                ApfloatRuntimeException

Bessel function of the second kind.

Parameters:: \u03bd - The order.; z - The argument.
Returns:: Y_ν(z)
Throws:: java.lang.ArithmeticException - If z is zero.; ApfloatRuntimeException
Since:: 1.13.0

besselK

public static Apcomplex besselK(Apcomplex ν,
                                Apcomplex z)
                         throws java.lang.ArithmeticException,
                                ApfloatRuntimeException

Modified Bessel function of the second kind.

Parameters:: \u03bd - The order.; z - The argument.
Returns:: K_ν(z)
Throws:: java.lang.ArithmeticException - If z is zero.; ApfloatRuntimeException
Since:: 1.13.0

ellipticK
```
public static Apcomplex ellipticK(Apcomplex z)
                           throws java.lang.ArithmeticException,
                                  ApfloatRuntimeException
```
Complete elliptic integral of the first kind.
Note that this function uses the definition: $K (z) ⩵ \int_{0}^{\frac{π}{2}} \frac{1}{\sqrt{1 - z \sin^{2} (t)}} ⅆ t$

Parameters:

z - The argument.

Returns:

K(z)

Throws:

InfiniteExpansionException - If z is zero.

java.lang.ArithmeticException - If z is one.

ApfloatRuntimeException

Since:

1.13.0

ellipticK

static Apcomplex ellipticK(Apcomplex z,
                           long precision)
                    throws java.lang.ArithmeticException,
                           ApfloatRuntimeException

Throws:: java.lang.ArithmeticException; ApfloatRuntimeException

ellipticK

static Apcomplex ellipticK(Apcomplex z,
                           long precision,
                           java.util.function.Consumer<Apcomplex> consumer)
                    throws java.lang.ArithmeticException,
                           ApfloatRuntimeException

Throws:: java.lang.ArithmeticException; ApfloatRuntimeException

ellipticE
```
public static Apcomplex ellipticE(Apcomplex z)
                           throws ApfloatRuntimeException
```
Complete elliptic integral of the second kind.
Note that this function uses the definition: $E (z) ⩵ \int_{0}^{\frac{π}{2}} \sqrt{1 - z \sin^{2} (t)} ⅆ t$

Parameters:

z - The argument.

Returns:

E(z)

Throws:

InfiniteExpansionException - If z is zero.

ApfloatRuntimeException

Since:

1.13.0

ellipticE

static Apcomplex ellipticE(Apcomplex z,
                           long precision)
                    throws ApfloatRuntimeException

Throws:: ApfloatRuntimeException

hermiteH

public static Apcomplex hermiteH(Apcomplex ν,
                                 Apcomplex z)
                          throws ApfloatRuntimeException

Hermite function. For integer values of ν gives the Hermite polynomial.

Parameters:: \u03bd - The first argument.; z - The second argument.
Returns:: H_ν(z)
Throws:: ApfloatRuntimeException
Since:: 1.14.0

laguerreL

public static Apcomplex laguerreL(Apcomplex ν,
                                  Apcomplex z)
                           throws ApfloatRuntimeException

Laguerre function. For integer values of ν gives the Laguerre polynomial.

Parameters:: \u03bd - The first argument.; z - The second argument.
Returns:: L_ν(z)
Throws:: ApfloatRuntimeException
Since:: 1.14.0

laguerreL

public static Apcomplex laguerreL(Apcomplex ν,
                                  Apcomplex λ,
                                  Apcomplex z)
                           throws ApfloatRuntimeException

Generalized Laguerre function. For integer values of ν gives the generalized Laguerre polynomial.

Parameters:: \u03bd - The first argument.; \u03bb - The second argument.; z - The third argument.
Returns:: L_ν^λ(z)
Throws:: ApfloatRuntimeException
Since:: 1.14.0

legendreP

public static Apcomplex legendreP(Apcomplex ν,
                                  Apcomplex z)
                           throws java.lang.ArithmeticException,
                                  ApfloatRuntimeException

Legendre function. For integer values of ν gives the Legendre polynomial.

Parameters:: \u03bd - The first argument.; z - The second argument.
Returns:: P_ν(z)
Throws:: java.lang.ArithmeticException - If ν is not an integer and z is -1.; ApfloatRuntimeException
Since:: 1.14.0

legendreP

public static Apcomplex legendreP(Apcomplex ν,
                                  Apcomplex μ,
                                  Apcomplex z)
                           throws java.lang.ArithmeticException,
                                  ApfloatRuntimeException

Associated Legendre function of the first kind. Gives Legendre functions of type 2.

Parameters:: \u03bd - The first argument.; \u03bc - The second argument.; z - The third argument.
Returns:: P_ν^μ(z)
Throws:: java.lang.ArithmeticException - If ν is not an integer and z is -1.; ApfloatRuntimeException
Since:: 1.14.0

legendreQ

public static Apcomplex legendreQ(Apcomplex ν,
                                  Apcomplex z)
                           throws java.lang.ArithmeticException,
                                  ApfloatRuntimeException

Legendre function of the second kind.

Parameters:: \u03bd - The first argument.; z - The second argument.
Returns:: Q_ν(z)
Throws:: java.lang.ArithmeticException - If z is 1 or -1.; ApfloatRuntimeException
Since:: 1.14.0

legendreQ

public static Apcomplex legendreQ(Apcomplex ν,
                                  Apcomplex μ,
                                  Apcomplex z)
                           throws java.lang.ArithmeticException,
                                  ApfloatRuntimeException

Associated Legendre function of the second kind. Gives Legendre functions of type 2.

Parameters:: \u03bd - The first argument.; \u03bc - The second argument.; z - The third argument.
Returns:: Q_ν^μ(z)
Throws:: java.lang.ArithmeticException - If z is 1 or -1.; ApfloatRuntimeException
Since:: 1.14.0

sphericalHarmonicY

public static Apcomplex sphericalHarmonicY(Apcomplex λ,
                                           Apcomplex μ,
                                           Apcomplex ϑ,
                                           Apcomplex ϕ)
                                    throws java.lang.ArithmeticException,
                                           ApfloatRuntimeException

Spherical harmonic function.

Parameters:: \u03bb - The first argument.; \u03bc - The second argument.; \u03d1 - The third argument.; \u03d5 - The fourth argument.
Returns:: Y_λ^μ(ϑ, φ)
Throws:: java.lang.ArithmeticException - If ϑ is π plus a multiple of 2 π and μ is not an integer and has a negative real part, or if λ - μ is a negative integer.; ApfloatRuntimeException
Since:: 1.14.0

sphericalHarmonicY

private static Apcomplex sphericalHarmonicY(Apint n,
                                            Apint m,
                                            Apcomplex ϑ,
                                            Apcomplex ϕ)
                                     throws java.lang.ArithmeticException,
                                            ApfloatRuntimeException

Throws:: java.lang.ArithmeticException; ApfloatRuntimeException

chebyshevT
```
public static Apcomplex chebyshevT(Apcomplex ν,
                                   Apcomplex z)
                            throws ApfloatRuntimeException
```
Chebyshev function of the first kind. For integer values of ν gives the Chebyshev polynomial of the first kind.

Parameters:

\u03bd - The first argument.

z - The second argument.

Returns:

T_ν(z)

Throws:

ApfloatRuntimeException

Since:

1.14.0

chebyshevU

public static Apcomplex chebyshevU(Apcomplex ν,
                                   Apcomplex z)
                            throws java.lang.ArithmeticException,
                                   ApfloatRuntimeException

Chebyshev function of the second kind. For integer values of ν gives the Chebyshev polynomial of the second kind.

Parameters:: \u03bd - The first argument.; z - The second argument.
Returns:: U_ν(z)
Throws:: java.lang.ArithmeticException - If z is -1 and ν is not an integer.; ApfloatRuntimeException
Since:: 1.14.0

gegenbauerC

public static Apcomplex gegenbauerC(Apcomplex ν,
                                    Apcomplex z)
                             throws java.lang.ArithmeticException,
                                    ApfloatRuntimeException

Renormalized Gegenbauer function.

Parameters:: \u03bd - The first argument.; z - The second argument.
Returns:: C_ν⁽⁰⁾(z)
Throws:: java.lang.ArithmeticException - If ν is zero.; ApfloatRuntimeException
Since:: 1.14.0

gegenbauerC

public static Apcomplex gegenbauerC(Apcomplex ν,
                                    Apcomplex λ,
                                    Apcomplex z)
                             throws java.lang.ArithmeticException,
                                    ApfloatRuntimeException

Gegenbauer function. For nonnegative integer values of ν gives the Gegenbauer polynomial.

Parameters:: \u03bd - The first argument.; \u03bb - The second argument.; z - The third argument.
Returns:: C_ν^λ(z)
Throws:: java.lang.ArithmeticException - If z is -1 and real part of λ is > 1/2. Also if z is -1 and λ is 1/2 and ν is not an integer.; ApfloatRuntimeException
Since:: 1.14.0

gegenbauerC

private static Apcomplex gegenbauerC(long n,
                                     Apcomplex λ,
                                     Apcomplex z)
                              throws java.lang.ArithmeticException,
                                     ApfloatRuntimeException

Throws:: java.lang.ArithmeticException; ApfloatRuntimeException

jacobiP

public static Apcomplex jacobiP(Apcomplex ν,
                                Apcomplex a,
                                Apcomplex b,
                                Apcomplex z)
                         throws java.lang.ArithmeticException,
                                ApfloatRuntimeException

Jacobi function. For nonnegative integer values of ν gives the Jacobi polynomial.

Parameters:: \u03bd - The first argument.; a - The second argument.; b - The third argument.; z - The fourth argument.
Returns:: P_ν^(a,b)(z)
Throws:: java.lang.ArithmeticException - If z is -1 and real part of b is > 0 and ν is not a positive integer. Also if ν + a is a negative integer and ν is not an integer.; ApfloatRuntimeException
Since:: 1.14.0

jacobiP

private static Apcomplex jacobiP(long n,
                                 Apcomplex a,
                                 Apcomplex b,
                                 Apcomplex z)
                          throws java.lang.ArithmeticException,
                                 ApfloatRuntimeException

Throws:: java.lang.ArithmeticException; ApfloatRuntimeException

fibonacci

public static Apcomplex fibonacci(Apcomplex ν,
                                  Apcomplex z)
                           throws java.lang.ArithmeticException,
                                  ApfloatRuntimeException

Fibonacci function. For nonnegative integer values of ν gives the Fibonacci polynomial.

Parameters:: \u03bd - The first argument.; z - The second argument.
Returns:: F_ν(z)
Throws:: java.lang.ArithmeticException - If z is -1 and ν is not an integer.; ApfloatRuntimeException
Since:: 1.14.0

eulerE

public static Apcomplex eulerE(long n,
                               Apcomplex z)
                        throws java.lang.IllegalArgumentException,
                               ApfloatRuntimeException

Euler polynomial.

Parameters:: n - The first argument.; z - The second argument.
Returns:: E_n(z)
Throws:: java.lang.IllegalArgumentException - If n < 0.; ApfloatRuntimeException
Since:: 1.14.0

eulerE

static Apcomplex eulerE(long n,
                        Apcomplex z,
                        long precision)
                 throws java.lang.IllegalArgumentException,
                        ApfloatRuntimeException

Throws:: java.lang.IllegalArgumentException; ApfloatRuntimeException

bernoulliB

public static Apcomplex bernoulliB(long n,
                                   Apcomplex z)
                            throws java.lang.IllegalArgumentException,
                                   ApfloatRuntimeException

Bernoulli polynomial.

Parameters:: n - The first argument.; z - The second argument.
Returns:: B_n(z)
Throws:: java.lang.IllegalArgumentException - If n < 0.; ApfloatRuntimeException
Since:: 1.14.0

bernoulliB

static Apcomplex bernoulliB(long n,
                            Apcomplex z,
                            long precision)
                     throws java.lang.IllegalArgumentException,
                            ApfloatRuntimeException

Throws:: java.lang.IllegalArgumentException; ApfloatRuntimeException

harmonicNumber

public static Apcomplex harmonicNumber(Apcomplex z)
                                throws java.lang.ArithmeticException,
                                       ApfloatRuntimeException

Harmonic number.

Parameters:: z - The argument.
Returns:: H_z
Throws:: java.lang.ArithmeticException - If z is a negative integer.; ApfloatRuntimeException
Since:: 1.14.0

harmonicNumber

public static Apcomplex harmonicNumber(Apcomplex z,
                                       Apcomplex r)
                                throws java.lang.ArithmeticException,
                                       ApfloatRuntimeException

Generalized harmonic number.

Parameters:: z - The first argument.; r - The second argument.
Returns:: H_z^(r)
Throws:: java.lang.ArithmeticException - If z is a negative integer, unless r has a negative real part or is zero.; ApfloatRuntimeException
Since:: 1.14.0

polylog

public static Apcomplex polylog(Apcomplex ν,
                                Apcomplex z)
                         throws java.lang.ArithmeticException,
                                ApfloatRuntimeException

Polylogarithm.

Parameters:: \u03bd - The first argument.; z - The second argument.
Returns:: Li_ν(z)
Throws:: java.lang.ArithmeticException - If the real part of ν is ≤ 1 and z is 1.; ApfloatRuntimeException
Since:: 1.14.0

logisticSigmoid

public static Apcomplex logisticSigmoid(Apcomplex z)
                                 throws java.lang.ArithmeticException,
                                        ApfloatRuntimeException

Logistic sigmoid.

Parameters:: z - The argument.
Returns:: σ(z)
Throws:: java.lang.ArithmeticException - If z is an odd integer multiple of π i.; ApfloatRuntimeException
Since:: 1.14.0

ulp
```
public static Apfloat ulp(Apcomplex z)
```
Returns the unit in the last place of the argument, considering the scale and precision. This is maximum of the ulps of the real and imaginary part of the argument. If the precision of the argument is infinite, zero is returned.

Parameters:

z - The argument.

Returns:

The ulp of the argument.

Since:

1.10.0

lastIterationExtendPrecision

private static Apcomplex lastIterationExtendPrecision(int iterations,
                                                      int precisingIteration,
                                                      Apcomplex z)

isNonPositiveInteger

static boolean isNonPositiveInteger(Apcomplex z)

Class ApcomplexMath

Constructor Summary

Method Summary

Methods inherited from class java.lang.Object

Constructor Detail

ApcomplexMath

Method Detail

negate

abs

norm

arg

scale

pow

powAbs

sqrt

cbrt

root

root

inverseRoot

inverseRoot

inverseRootAbs

allRoots

agm

agm

limitPrecision

rightSqrt

agmConsume

log

log

rawLog

exp

pow

acos

acos

acosh

acosh

asin

asinh

atan

atanh

cos

cosh

sin

sinh

tan

tanFixedPrecision

tan

tanh

tanhFixedPrecision

tanh

cot

sinc

w

w

product

sum

gamma

gamma

gamma

logGamma

logSin

expNoLoP

logPochhammer

digamma

polygamma

beta

beta

beta

pochhammer

pochhammer

binomial

zeta

zeta

hypergeometric0F1

hypergeometric0F1Regularized

hypergeometric1F1

hypergeometric1F1Regularized

hypergeometric2F1

hypergeometric2F1Regularized

hypergeometricU