java.lang.Object

org.apfloat.ApcomplexMath

public class ApcomplexMath extends Object

Various mathematical functions for arbitrary precision complex numbers.

Version:

1.14.0

Author:

Mikko Tommila

See Also:

Method Summary

Modifier and Type

Method

Description

static Apfloat

abs(Apcomplex z)

Absolute value.

static Apcomplex

acos(Apcomplex z)

Inverse cosine.

static Apcomplex

acosh(Apcomplex z)

Inverse hyperbolic cosine.

static Apcomplex

agm(Apcomplex a, Apcomplex b)

Arithmetic-geometric mean.

static Apcomplex

airyAi(Apcomplex z)

Airy function Ai.

static Apcomplex

airyAiPrime(Apcomplex z)

Derivative of the Airy function Ai.

static Apcomplex

airyBi(Apcomplex z)

Airy function Bi.

static Apcomplex

airyBiPrime(Apcomplex z)

Derivative of the Airy function Bi.

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.

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.

static Apcomplex

digamma(Apcomplex z)

Digamma function.

static Apcomplex

ellipticE(Apcomplex z)

Complete elliptic integral of the second kind.

static Apcomplex

ellipticK(Apcomplex z)

Complete elliptic integral of the first kind.

static Apcomplex

erf(Apcomplex z)

Error function.

static Apcomplex

erfc(Apcomplex z)

Complementary error function.

static Apcomplex

erfi(Apcomplex z)

Imaginary error function.

static Apcomplex

eulerE(long n, Apcomplex z)

Euler polynomial.

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.

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.

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.

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.

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.

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.

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.

static Apcomplex

negate(Apcomplex z)

Deprecated.
Use Apcomplex.negate().

static Apfloat

norm(Apcomplex z)

Norm.

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.

static Apcomplex

product(Apcomplex... z)

Product of numbers.

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.

static Apcomplex

sqrt(Apcomplex z)

Square root.

static Apcomplex

sum(Apcomplex... z)

Sum of numbers.

static Apcomplex

tan(Apcomplex z)

Tangent.

static Apcomplex

tanh(Apcomplex z)

Hyperbolic tangent.

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

Method Details
- 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 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:
  
  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 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:
  
  ArithmeticException - If both z and n are zero.
  
  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 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:
  
  ArithmeticException - If n is zero.
  
  ApfloatRuntimeException
- root
  
  public static Apcomplex root(Apcomplex z, long n, long k) throws 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:
  
  ArithmeticException - If n is zero.
  
  ApfloatRuntimeException
  
  Since:
  
  1.5
- inverseRoot
  
  public static Apcomplex inverseRoot(Apcomplex z, long n) throws 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:
  
  ArithmeticException - If z or n is zero.
  
  ApfloatRuntimeException
- inverseRoot
  
  public static Apcomplex inverseRoot(Apcomplex z, long n, long k) throws 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:
  
  ArithmeticException - If z or n is zero.
  
  ApfloatRuntimeException
- allRoots
  
  public static Apcomplex[] allRoots(Apcomplex z, int n) throws 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:
  
  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
- log
  
  public static Apcomplex log(Apcomplex z) throws 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:
  
  ArithmeticException - If z is zero.
  
  ApfloatRuntimeException
- log
  
  public static Apcomplex log(Apcomplex z, Apcomplex w) throws ArithmeticException, ApfloatRuntimeException
  
  Logarithm in arbitrary base.
  
  Parameters:
  
  z - The argument.
  
  w - The base.
  
  Returns:
  
  Base-w logarithm of z.
  
  Throws:
  
  ArithmeticException - If z or w is zero.
  
  ApfloatRuntimeException
  
  Since:
  
  1.6
- 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:
  
  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
- 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
- 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 ArithmeticException, ApfloatRuntimeException
  
  Inverse tangent. Calculated using log().
  
  Parameters:
  
  z - The argument.
  
  Returns:
  
  Inverse tangent of z.
  
  Throws:
  
  ArithmeticException - If z == i.
  
  ApfloatRuntimeException
- atanh
  
  public static Apcomplex atanh(Apcomplex z) throws ArithmeticException, ApfloatRuntimeException
  
  Inverse hyperbolic tangent. Calculated using log().
  
  Parameters:
  
  z - The argument.
  
  Returns:
  
  Inverse hyperbolic tangent of z.
  
  Throws:
  
  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 ArithmeticException, ApfloatRuntimeException
  
  Tangent. Calculated using exp().
  
  Parameters:
  
  z - The argument.
  
  Returns:
  
  Tangent of z.
  
  Throws:
  
  ArithmeticException - If z is π/2 + n π where n is an integer.
  
  ApfloatRuntimeException
- tanh
  
  public static Apcomplex tanh(Apcomplex z) throws ArithmeticException, ApfloatRuntimeException
  
  Hyperbolic tangent. Calculated using exp().
  
  Parameters:
  
  z - The argument.
  
  Returns:
  
  Hyperbolic tangent of z.
  
  Throws:
  
  ArithmeticException - If z is i (π/2 + n π) where n is an integer.
  
  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 ArithmeticException, ApfloatRuntimeException
  
  Lambert W function for the specified branch.
  
  Parameters:
  
  z - The argument.
  
  k - The branch.
  
  Returns:
  
  W_k(z).
  
  Throws:
  
  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 ArithmeticException, ApfloatRuntimeException
  
  Gamma function.
  
  Parameters:
  
  z - The argument.
  
  Returns:
  
  Γ(z)
  
  Throws:
  
  ArithmeticException - If z is a nonpositive integer.
  
  ApfloatRuntimeException
  
  Since:
  
  1.9.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. The asymptotic complexity is at least O(n²log n) and it is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the gamma function.
- gamma
  
  public static Apcomplex gamma(Apcomplex a, Apcomplex z) throws ArithmeticException, ApfloatRuntimeException
  
  Incomplete gamma function.
  
  Parameters:
  
  a - The first argument.
  
  z - The second argument.
  
  Returns:
  
  Γ(a, z)
  
  Throws:
  
  ArithmeticException - If the real part of a is nonpositive and z is zero.
  
  ApfloatRuntimeException
  
  Since:
  
  1.10.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. The asymptotic complexity is at least O(n²log n) and it is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the incomplete gamma function.
- gamma
  
  public static Apcomplex gamma(Apcomplex a, Apcomplex z0, Apcomplex z1) throws 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:
  
  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
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. The asymptotic complexity is at least O(n²log n) and it is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the incomplete gamma function.
- logGamma
  
  public static Apcomplex logGamma(Apcomplex z) throws 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:
  
  ArithmeticException - If z is a nonpositive integer.
  
  ApfloatRuntimeException
  
  Since:
  
  1.11.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. The asymptotic complexity is at least O(n²log n) and it is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the gamma function.
- digamma
  
  public static Apcomplex digamma(Apcomplex z) throws ArithmeticException, ApfloatRuntimeException
  
  Digamma function.
  
  Parameters:
  
  z - The argument.
  
  Returns:
  
  ψ(z)
  
  Throws:
  
  ArithmeticException - If z is a nonpositive integer.
  
  ApfloatRuntimeException
  
  Since:
  
  1.11.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. The asymptotic complexity is at least O(n²log n) and it is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the digamma function.
- polygamma
  
  public static Apcomplex polygamma(long n, Apcomplex z) throws ArithmeticException, ApfloatRuntimeException
  
  Polygamma function.
  
  Parameters:
  
  n - The order.
  
  z - The argument.
  
  Returns:
  
  ψ⁽ⁿ⁾(z)
  
  Throws:
  
  ArithmeticException - If n is negative or z is a nonpositive integer.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. The asymptotic complexity is at least O(n²log n) and it is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the polygamma function.
- beta
  
  public static Apcomplex beta(Apcomplex a, Apcomplex b) throws ArithmeticException, ApfloatRuntimeException
  
  Beta function.
  
  Parameters:
  
  a - The first argument.
  
  b - The second argument.
  
  Returns:
  
  B(a, b)
  
  Throws:
  
  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
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. The asymptotic complexity is at least O(n²log n) and it is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the beta function.
- beta
  
  public static Apcomplex beta(Apcomplex z, Apcomplex a, Apcomplex b) throws ArithmeticException, ApfloatRuntimeException
  
  Incomplete beta function.
  
  Parameters:
  
  z - The first argument.
  
  a - The second argument.
  
  b - The third argument.
  
  Returns:
  
  B_z(a, b)
  
  Throws:
  
  ArithmeticException - If a is a nonpositive integer or z is zero and a has nonpositive real part.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. The asymptotic complexity is at least O(n²log n) and it is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- beta
  
  public static Apcomplex beta(Apcomplex z1, Apcomplex z2, Apcomplex a, Apcomplex b) throws 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:
  
  ArithmeticException - If a is a nonpositive integer or z1 or z2 is zero and a has nonpositive real part.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. The asymptotic complexity is at least O(n²log n) and it is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- pochhammer
  
  public static Apcomplex pochhammer(Apcomplex z, Apcomplex n) throws ArithmeticException, ApfloatRuntimeException
  
  Pochhammer symbol.
  
  Parameters:
  
  z - The first argument.
  
  n - The second argument.
  
  Returns:
  
  (z)_n
  
  Throws:
  
  ArithmeticException - If z + n is a nonpositive integer but z is not.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. The asymptotic complexity is at least O(n²log n) and it is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the pochhammer symbol.
- binomial
  
  public static Apcomplex binomial(Apcomplex n, Apcomplex k) throws ArithmeticException, ApfloatRuntimeException
  
  Binomial coefficient. Calculated using the gamma(Apcomplex) function.
  
  Parameters:
  
  n - The first argument.
  
  k - The second argument.
  
  Returns:
  
  $(\binom{n}{k})$
  
  Throws:
  
  ArithmeticException - If n is a negative integer and k is noninteger.
  
  ApfloatRuntimeException
  
  Since:
  
  1.11.0
- zeta
  
  public static Apcomplex zeta(Apcomplex s) throws ArithmeticException, ApfloatRuntimeException
  
  Riemann zeta function.
  
  Parameters:
  
  s - The argument.
  
  Returns:
  
  ζ(s)
  
  Throws:
  
  ArithmeticException - If s is 1.
  
  ApfloatRuntimeException
  
  Since:
  
  1.11.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few hundred digits. At the time of implementation no generic fast algorithm is known for the zeta function.
- zeta
  
  public static Apcomplex zeta(Apcomplex s, Apcomplex a) throws ArithmeticException, ApfloatRuntimeException
  
  Hurwitz zeta function.
  
  Parameters:
  
  s - The first argument.
  
  a - The second argument.
  
  Returns:
  
  ζ(s, a)
  
  Throws:
  
  ArithmeticException - If s is 1 or if a is a nonpositive integer.
  
  ApfloatRuntimeException
  
  Since:
  
  1.11.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few hundred digits. At the time of implementation no generic fast algorithm is known for the zeta function.
- hypergeometric0F1
  
  public static Apcomplex hypergeometric0F1(Apcomplex a, Apcomplex z) throws ArithmeticException, ApfloatRuntimeException
  
  Confluent hypergeometric function ₀F₁.
  
  Parameters:
  
  a - The first argument.
  
  z - The second argument.
  
  Returns:
  
  ₀F₁(; a; z)
  
  Throws:
  
  ArithmeticException - If the function value is not finite.
  
  ApfloatRuntimeException
  
  Since:
  
  1.11.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- 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
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- hypergeometric1F1
  
  public static Apcomplex hypergeometric1F1(Apcomplex a, Apcomplex b, Apcomplex z) throws 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:
  
  ArithmeticException - If the function value is not finite.
  
  ApfloatRuntimeException
  
  Since:
  
  1.11.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- 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
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- hypergeometric2F1
  
  public static Apcomplex hypergeometric2F1(Apcomplex a, Apcomplex b, Apcomplex c, Apcomplex z) throws 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:
  
  ArithmeticException - If the function value is not finite.
  
  ApfloatRuntimeException
  
  Since:
  
  1.11.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- 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
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- hypergeometricU
  
  public static Apcomplex hypergeometricU(Apcomplex a, Apcomplex b, Apcomplex z) throws 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:
  
  ArithmeticException - If the result is not finite.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- erf
  
  public static Apcomplex erf(Apcomplex z) throws ApfloatRuntimeException
  
  Error function.
  
  Parameters:
  
  z - The argument.
  
  Returns:
  
  erf(z)
  
  Throws:
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- 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
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- 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
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- 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
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- 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
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- expIntegralE
  
  public static Apcomplex expIntegralE(Apcomplex ν, Apcomplex z) throws ArithmeticException, ApfloatRuntimeException
  
  Exponential integral E.
  
  Parameters:
  
  ν - The first argument.
  
  z - The second argument.
  
  Returns:
  
  E_ν(z)
  
  Throws:
  
  ArithmeticException - If real part of ν is ≤ 1 and z is zero.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- expIntegralEi
  
  public static Apcomplex expIntegralEi(Apcomplex z) throws ArithmeticException, ApfloatRuntimeException
  
  Exponential integral Ei.
  
  Parameters:
  
  z - The argument.
  
  Returns:
  
  Ei(z)
  
  Throws:
  
  ArithmeticException - If z is zero.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- logIntegral
  
  public static Apcomplex logIntegral(Apcomplex z) throws ArithmeticException, ApfloatRuntimeException
  
  Logarithmic integral.
  
  Parameters:
  
  z - The argument.
  
  Returns:
  
  li(z)
  
  Throws:
  
  ArithmeticException - If z is zero.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- sinIntegral
  
  public static Apcomplex sinIntegral(Apcomplex z) throws ApfloatRuntimeException
  
  Sine integral.
  
  Parameters:
  
  z - The argument.
  
  Returns:
  
  Si(z)
  
  Throws:
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- cosIntegral
  
  public static Apcomplex cosIntegral(Apcomplex z) throws ArithmeticException, ApfloatRuntimeException
  
  Cosine integral.
  
  Parameters:
  
  z - The argument.
  
  Returns:
  
  Ci(z)
  
  Throws:
  
  ArithmeticException - If z is zero.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- 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
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- coshIntegral
  
  public static Apcomplex coshIntegral(Apcomplex z) throws ArithmeticException, ApfloatRuntimeException
  
  Hyperbolic cosine integral.
  
  Parameters:
  
  z - The argument.
  
  Returns:
  
  Chi(z)
  
  Throws:
  
  ArithmeticException - If z is zero.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- 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
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- 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
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- 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
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- 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
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- besselJ
  
  public static Apcomplex besselJ(Apcomplex ν, Apcomplex z) throws ArithmeticException, ApfloatRuntimeException
  
  Bessel function of the first kind.
  
  Parameters:
  
  ν - The order.
  
  z - The argument.
  
  Returns:
  
  J_ν(z)
  
  Throws:
  
  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
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- besselI
  
  public static Apcomplex besselI(Apcomplex ν, Apcomplex z) throws ArithmeticException, ApfloatRuntimeException
  
  Modified Bessel function of the first kind.
  
  Parameters:
  
  ν - The order.
  
  z - The argument.
  
  Returns:
  
  I_ν(z)
  
  Throws:
  
  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
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- besselY
  
  public static Apcomplex besselY(Apcomplex ν, Apcomplex z) throws ArithmeticException, ApfloatRuntimeException
  
  Bessel function of the second kind.
  
  Parameters:
  
  ν - The order.
  
  z - The argument.
  
  Returns:
  
  Y_ν(z)
  
  Throws:
  
  ArithmeticException - If z is zero.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- besselK
  
  public static Apcomplex besselK(Apcomplex ν, Apcomplex z) throws ArithmeticException, ApfloatRuntimeException
  
  Modified Bessel function of the second kind.
  
  Parameters:
  
  ν - The order.
  
  z - The argument.
  
  Returns:
  
  K_ν(z)
  
  Throws:
  
  ArithmeticException - If z is zero.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- ellipticK
  
  public static Apcomplex ellipticK(Apcomplex z) throws 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.
  
  ArithmeticException - If z is one.
  
  ApfloatRuntimeException
  
  Since:
  
  1.13.0
- 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
- hermiteH
  
  public static Apcomplex hermiteH(Apcomplex ν, Apcomplex z) throws ApfloatRuntimeException
  
  Hermite function. For integer values of ν gives the Hermite polynomial.
  
  Parameters:
  
  ν - The first argument.
  
  z - The second argument.
  
  Returns:
  
  H_ν(z)
  
  Throws:
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- laguerreL
  
  public static Apcomplex laguerreL(Apcomplex ν, Apcomplex z) throws ApfloatRuntimeException
  
  Laguerre function. For integer values of ν gives the Laguerre polynomial.
  
  Parameters:
  
  ν - The first argument.
  
  z - The second argument.
  
  Returns:
  
  L_ν(z)
  
  Throws:
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- 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:
  
  ν - The first argument.
  
  λ - The second argument.
  
  z - The third argument.
  
  Returns:
  
  L_ν^λ(z)
  
  Throws:
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- legendreP
  
  public static Apcomplex legendreP(Apcomplex ν, Apcomplex z) throws ArithmeticException, ApfloatRuntimeException
  
  Legendre function. For integer values of ν gives the Legendre polynomial.
  
  Parameters:
  
  ν - The first argument.
  
  z - The second argument.
  
  Returns:
  
  P_ν(z)
  
  Throws:
  
  ArithmeticException - If ν is not an integer and z is -1.
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- legendreP
  
  public static Apcomplex legendreP(Apcomplex ν, Apcomplex μ, Apcomplex z) throws ArithmeticException, ApfloatRuntimeException
  
  Associated Legendre function of the first kind. Gives Legendre functions of type 2.
  
  Parameters:
  
  ν - The first argument.
  
  μ - The second argument.
  
  z - The third argument.
  
  Returns:
  
  P_ν^μ(z)
  
  Throws:
  
  ArithmeticException - If ν is not an integer and z is -1.
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- legendreQ
  
  public static Apcomplex legendreQ(Apcomplex ν, Apcomplex z) throws ArithmeticException, ApfloatRuntimeException
  
  Legendre function of the second kind.
  
  Parameters:
  
  ν - The first argument.
  
  z - The second argument.
  
  Returns:
  
  Q_ν(z)
  
  Throws:
  
  ArithmeticException - If z is 1 or -1.
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- legendreQ
  
  public static Apcomplex legendreQ(Apcomplex ν, Apcomplex μ, Apcomplex z) throws ArithmeticException, ApfloatRuntimeException
  
  Associated Legendre function of the second kind. Gives Legendre functions of type 2.
  
  Parameters:
  
  ν - The first argument.
  
  μ - The second argument.
  
  z - The third argument.
  
  Returns:
  
  Q_ν^μ(z)
  
  Throws:
  
  ArithmeticException - If z is 1 or -1.
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- sphericalHarmonicY
  
  public static Apcomplex sphericalHarmonicY(Apcomplex λ, Apcomplex μ, Apcomplex ϑ, Apcomplex ϕ) throws ArithmeticException, ApfloatRuntimeException
  
  Spherical harmonic function.
  
  Parameters:
  
  λ - The first argument.
  
  μ - The second argument.
  
  ϑ - The third argument.
  
  ϕ - The fourth argument.
  
  Returns:
  
  Y_λ^μ(ϑ, φ)
  
  Throws:
  
  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
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- 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:
  
  ν - 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 ArithmeticException, ApfloatRuntimeException
  
  Chebyshev function of the second kind. For integer values of ν gives the Chebyshev polynomial of the second kind.
  
  Parameters:
  
  ν - The first argument.
  
  z - The second argument.
  
  Returns:
  
  U_ν(z)
  
  Throws:
  
  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 ArithmeticException, ApfloatRuntimeException
  
  Renormalized Gegenbauer function.
  
  Parameters:
  
  ν - The first argument.
  
  z - The second argument.
  
  Returns:
  
  C_ν⁽⁰⁾(z)
  
  Throws:
  
  ArithmeticException - If ν is zero.
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
- gegenbauerC
  
  public static Apcomplex gegenbauerC(Apcomplex ν, Apcomplex λ, Apcomplex z) throws ArithmeticException, ApfloatRuntimeException
  
  Gegenbauer function. For nonnegative integer values of ν gives the Gegenbauer polynomial.
  
  Parameters:
  
  ν - The first argument.
  
  λ - The second argument.
  
  z - The third argument.
  
  Returns:
  
  C_ν^λ(z)
  
  Throws:
  
  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
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- jacobiP
  
  public static Apcomplex jacobiP(Apcomplex ν, Apcomplex a, Apcomplex b, Apcomplex z) throws ArithmeticException, ApfloatRuntimeException
  
  Jacobi function. For nonnegative integer values of ν gives the Jacobi polynomial.
  
  Parameters:
  
  ν - The first argument.
  
  a - The second argument.
  
  b - The third argument.
  
  z - The fourth argument.
  
  Returns:
  
  P_ν^(a,b)(z)
  
  Throws:
  
  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
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- fibonacci
  
  public static Apcomplex fibonacci(Apcomplex ν, Apcomplex z) throws ArithmeticException, ApfloatRuntimeException
  
  Fibonacci function. For nonnegative integer values of ν gives the Fibonacci polynomial.
  
  Parameters:
  
  ν - The first argument.
  
  z - The second argument.
  
  Returns:
  
  F_ν(z)
  
  Throws:
  
  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 IllegalArgumentException, ApfloatRuntimeException
  
  Euler polynomial.
  
  Parameters:
  
  n - The first argument.
  
  z - The second argument.
  
  Returns:
  
  E_n(z)
  
  Throws:
  
  IllegalArgumentException - If n < 0.
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- bernoulliB
  
  public static Apcomplex bernoulliB(long n, Apcomplex z) throws IllegalArgumentException, ApfloatRuntimeException
  
  Bernoulli polynomial.
  
  Parameters:
  
  n - The first argument.
  
  z - The second argument.
  
  Returns:
  
  B_n(z)
  
  Throws:
  
  IllegalArgumentException - If n < 0.
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- harmonicNumber
  
  public static Apcomplex harmonicNumber(Apcomplex z) throws ArithmeticException, ApfloatRuntimeException
  
  Harmonic number.
  
  Parameters:
  
  z - The argument.
  
  Returns:
  
  H_z
  
  Throws:
  
  ArithmeticException - If z is a negative integer.
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- harmonicNumber
  
  public static Apcomplex harmonicNumber(Apcomplex z, Apcomplex r) throws ArithmeticException, ApfloatRuntimeException
  
  Generalized harmonic number.
  
  Parameters:
  
  z - The first argument.
  
  r - The second argument.
  
  Returns:
  
  H_z^(r)
  
  Throws:
  
  ArithmeticException - If z is a negative integer, unless r has a negative real part or is zero.
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- polylog
  
  public static Apcomplex polylog(Apcomplex ν, Apcomplex z) throws ArithmeticException, ApfloatRuntimeException
  
  Polylogarithm.
  
  Parameters:
  
  ν - The first argument.
  
  z - The second argument.
  
  Returns:
  
  Li_ν(z)
  
  Throws:
  
  ArithmeticException - If the real part of ν is ≤ 1 and z is 1.
  
  ApfloatRuntimeException
  
  Since:
  
  1.14.0
  
  Implementation notes:
  
  This implementation is slow, meaning that it isn't a fast algorithm. It is impractically slow beyond a precision of a few thousand digits. At the time of implementation no generic fast algorithm is known for the function.
- logisticSigmoid
  
  public static Apcomplex logisticSigmoid(Apcomplex z) throws ArithmeticException, ApfloatRuntimeException
  
  Logistic sigmoid.
  
  Parameters:
  
  z - The argument.
  
  Returns:
  
  σ(z)
  
  Throws:
  
  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

Class ApcomplexMath

Method Summary

Methods inherited from class java.lang.Object

Method Details

negate

abs

norm

arg

scale

pow

sqrt

cbrt

root

root

inverseRoot

inverseRoot

allRoots

agm

log

log

exp

pow

acos

acosh

asin

asinh

atan

atanh

cos

cosh

sin

sinh

tan

tanh

sinc

w

w

product

sum

gamma

gamma

gamma

logGamma

digamma

polygamma

beta

beta

beta

pochhammer

binomial

zeta

zeta

hypergeometric0F1

hypergeometric0F1Regularized

hypergeometric1F1

hypergeometric1F1Regularized

hypergeometric2F1

hypergeometric2F1Regularized

hypergeometricU

erf

erfc

erfi

fresnelS

fresnelC

expIntegralE

expIntegralEi

logIntegral

sinIntegral

cosIntegral

sinhIntegral

coshIntegral

airyAi

airyAiPrime

airyBi

airyBiPrime

besselJ

besselI

besselY

besselK

ellipticK