Package org.apfloat

Class ApcomplexMath

java.lang.Object

org.apfloat.ApcomplexMath

public class ApcomplexMath
extends Object

Various mathematical functions for arbitrary precision complex numbers.

Version:

1.13.0

Author:

Mikko Tommila

See Also:

• ApfloatMath

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	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	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	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	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	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	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	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	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	sinh(Apcomplex z)	Hyperbolic sine.
static Apcomplex	sinhIntegral(Apcomplex z)	Hyperbolic sine integral.
static Apcomplex	sinIntegral(Apcomplex z)	Sine integral.
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 z1/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 z1/nei2πsk/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 z1/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:

ez.

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:

zw.

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

w

public static Apcomplex w(Apcomplex z)
                   throws ApfloatRuntimeException

Lambert W function. The W function gives the solution to the equation W eW = 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:

Wk(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:

$\left(\genfrac{}{}{0ex}{}{n}{k}\right)$

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.

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.

Parameters:

z - The argument.

Returns:

E(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.

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