Package org.apfloat
Class ApcomplexMath
- java.lang.Object
-
- org.apfloat.ApcomplexMath
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public class ApcomplexMath extends java.lang.Object
Various mathematical functions for arbitrary precision complex numbers.- Version:
- 1.14.0
- See Also:
ApfloatMath
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-
Constructor Summary
Constructors Modifier Constructor Description private
ApcomplexMath()
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Method Summary
All Methods Static Methods Concrete Methods Deprecated Methods Modifier and Type Method Description static Apfloat
abs(Apcomplex z)
Absolute value.static Apcomplex
acos(Apcomplex z)
Inverse cosine.(package private) static Apcomplex
acos(Apcomplex z, long precision)
static Apcomplex
acosh(Apcomplex z)
Inverse hyperbolic cosine.(package private) static Apcomplex
acosh(Apcomplex z, long precision)
static Apcomplex
agm(Apcomplex a, Apcomplex b)
Arithmetic-geometric mean.(package private) static Apcomplex
agm(Apcomplex a, Apcomplex b, java.util.function.Consumer<Apcomplex> consumer)
private static Apcomplex
agmConsume(java.util.function.Consumer<Apcomplex> consumer, Apcomplex a, Apcomplex c2, long workingPrecision)
private static Apcomplex
airy(java.util.function.Function<java.lang.Long,Apcomplex> f, long targetPrecision, int radix)
static Apcomplex
airyAi(Apcomplex z)
Airy function Ai.(package private) static Apcomplex
airyAi(Apcomplex z0, long targetPrecision)
static Apcomplex
airyAiPrime(Apcomplex z)
Derivative of the Airy function Ai.(package private) static Apcomplex
airyAiPrime(Apcomplex z0, long targetPrecision)
static Apcomplex
airyBi(Apcomplex z)
Airy function Bi.(package private) static Apcomplex
airyBi(Apcomplex z0, long targetPrecision)
static Apcomplex
airyBiPrime(Apcomplex z)
Derivative of the Airy function Bi.(package private) static Apcomplex
airyBiPrime(Apcomplex z0, long targetPrecision)
static Apcomplex[]
allRoots(Apcomplex z, int n)
All values of the positive integer root.static Apfloat
arg(Apcomplex z)
Angle of the complex vector in the complex plane.static Apcomplex
asin(Apcomplex z)
Inverse sine.static Apcomplex
asinh(Apcomplex z)
Inverse hyperbolic sine.static Apcomplex
atan(Apcomplex z)
Inverse tangent.static Apcomplex
atanh(Apcomplex z)
Inverse hyperbolic tangent.static Apcomplex
bernoulliB(long n, Apcomplex z)
Bernoulli polynomial.(package private) static Apcomplex
bernoulliB(long n, Apcomplex z, long precision)
static Apcomplex
besselI(Apcomplex ν, Apcomplex z)
Modified Bessel function of the first kind.static Apcomplex
besselJ(Apcomplex ν, Apcomplex z)
Bessel function of the first kind.static Apcomplex
besselK(Apcomplex ν, Apcomplex z)
Modified Bessel function of the second kind.static Apcomplex
besselY(Apcomplex ν, Apcomplex z)
Bessel function of the second kind.static Apcomplex
beta(Apcomplex a, Apcomplex b)
Beta function.static Apcomplex
beta(Apcomplex z, Apcomplex a, Apcomplex b)
Incomplete beta function.static Apcomplex
beta(Apcomplex z1, Apcomplex z2, Apcomplex a, Apcomplex b)
Generalized incomplete beta function.static Apcomplex
binomial(Apcomplex n, Apcomplex k)
Binomial coefficient.static Apcomplex
cbrt(Apcomplex z)
Cube root.static Apcomplex
chebyshevT(Apcomplex ν, Apcomplex z)
Chebyshev function of the first kind.static Apcomplex
chebyshevU(Apcomplex ν, Apcomplex z)
Chebyshev function of the second kind.static Apcomplex
cos(Apcomplex z)
Cosine.static Apcomplex
cosh(Apcomplex z)
Hyperbolic cosine.static Apcomplex
coshIntegral(Apcomplex z)
Hyperbolic cosine integral.static Apcomplex
cosIntegral(Apcomplex z)
Cosine integral.(package private) static Apcomplex
cot(Apcomplex z)
static Apcomplex
digamma(Apcomplex z)
Digamma function.static Apcomplex
ellipticE(Apcomplex z)
Complete elliptic integral of the second kind.(package private) static Apcomplex
ellipticE(Apcomplex z, long precision)
static Apcomplex
ellipticK(Apcomplex z)
Complete elliptic integral of the first kind.(package private) static Apcomplex
ellipticK(Apcomplex z, long precision)
(package private) static Apcomplex
ellipticK(Apcomplex z, long precision, java.util.function.Consumer<Apcomplex> consumer)
static Apcomplex
erf(Apcomplex z)
Error function.static Apcomplex
erfc(Apcomplex z)
Complementary error function.(package private) static Apcomplex
erfcFixedPrecision(Apcomplex z)
(package private) static Apcomplex
erfFixedPrecision(Apcomplex z)
static Apcomplex
erfi(Apcomplex z)
Imaginary error function.(package private) static Apcomplex
erfiFixedPrecision(Apcomplex z)
static Apcomplex
eulerE(long n, Apcomplex z)
Euler polynomial.(package private) static Apcomplex
eulerE(long n, Apcomplex z, long precision)
static Apcomplex
exp(Apcomplex z)
Exponent function.static Apcomplex
expIntegralE(Apcomplex ν, Apcomplex z)
Exponential integral E.static Apcomplex
expIntegralEi(Apcomplex z)
Exponential integral Ei.private static Apcomplex
expNoLoP(Apcomplex z)
static Apcomplex
fibonacci(Apcomplex ν, Apcomplex z)
Fibonacci function.static Apcomplex
fresnelC(Apcomplex z)
Fresnel integral C.static Apcomplex
fresnelS(Apcomplex z)
Fresnel integral S.private static Apcomplex
fresnelTerm(Apint one, Apfloat half, Apfloat invSqrtPi, Apcomplex iz2, Apcomplex iHalfPiZ2)
static Apcomplex
gamma(Apcomplex z)
Gamma function.static Apcomplex
gamma(Apcomplex a, Apcomplex z)
Incomplete gamma function.static Apcomplex
gamma(Apcomplex a, Apcomplex z0, Apcomplex z1)
Generalized incomplete gamma function.private static Apcomplex
gegenbauerC(long n, Apcomplex λ, Apcomplex z)
static Apcomplex
gegenbauerC(Apcomplex ν, Apcomplex z)
Renormalized Gegenbauer function.static Apcomplex
gegenbauerC(Apcomplex ν, Apcomplex λ, Apcomplex z)
Gegenbauer function.static Apcomplex
harmonicNumber(Apcomplex z)
Harmonic number.static Apcomplex
harmonicNumber(Apcomplex z, Apcomplex r)
Generalized harmonic number.static Apcomplex
hermiteH(Apcomplex ν, Apcomplex z)
Hermite function.static Apcomplex
hypergeometric0F1(Apcomplex a, Apcomplex z)
Confluent hypergeometric function 0F1.static Apcomplex
hypergeometric0F1Regularized(Apcomplex a, Apcomplex z)
Regularized confluent hypergeometric function 0F̃1.static Apcomplex
hypergeometric1F1(Apcomplex a, Apcomplex b, Apcomplex z)
Kummer confluent hypergeometric function 1F1.static Apcomplex
hypergeometric1F1Regularized(Apcomplex a, Apcomplex b, Apcomplex z)
Regularized Kummer confluent hypergeometric function 1F̃1.static Apcomplex
hypergeometric2F1(Apcomplex a, Apcomplex b, Apcomplex c, Apcomplex z)
Hypergeometric function 2F1.static Apcomplex
hypergeometric2F1Regularized(Apcomplex a, Apcomplex b, Apcomplex c, Apcomplex z)
Regularized hypergeometric function 2F̃1.static Apcomplex
hypergeometricU(Apcomplex a, Apcomplex b, Apcomplex z)
Tricomi's confluent hypergeometric function U.static Apcomplex
inverseRoot(Apcomplex z, long n)
Inverse positive integer root.static Apcomplex
inverseRoot(Apcomplex z, long n, long k)
Inverse positive integer root.private static Apcomplex
inverseRootAbs(Apcomplex z, long n, long k)
(package private) static boolean
isNonPositiveInteger(Apcomplex z)
private static Apcomplex
jacobiP(long n, Apcomplex a, Apcomplex b, Apcomplex z)
static Apcomplex
jacobiP(Apcomplex ν, Apcomplex a, Apcomplex b, Apcomplex z)
Jacobi function.static Apcomplex
laguerreL(Apcomplex ν, Apcomplex z)
Laguerre function.static Apcomplex
laguerreL(Apcomplex ν, Apcomplex λ, Apcomplex z)
Generalized Laguerre function.private static Apcomplex
lastIterationExtendPrecision(int iterations, int precisingIteration, Apcomplex z)
static Apcomplex
legendreP(Apcomplex ν, Apcomplex z)
Legendre function.static Apcomplex
legendreP(Apcomplex ν, Apcomplex μ, Apcomplex z)
Associated Legendre function of the first kind.static Apcomplex
legendreQ(Apcomplex ν, Apcomplex z)
Legendre function of the second kind.static Apcomplex
legendreQ(Apcomplex ν, Apcomplex μ, Apcomplex z)
Associated Legendre function of the second kind.private static Apcomplex
limitPrecision(Apcomplex z)
static Apcomplex
log(Apcomplex z)
Natural logarithm.static Apcomplex
log(Apcomplex z, Apcomplex w)
Logarithm in arbitrary base.static Apcomplex
logGamma(Apcomplex z)
Logarithm of the gamma function.static Apcomplex
logIntegral(Apcomplex z)
Logarithmic integral.static Apcomplex
logisticSigmoid(Apcomplex z)
Logistic sigmoid.private static Apcomplex
logPochhammer(Apcomplex z, long n)
private static Apcomplex
logSin(Apcomplex z)
static Apcomplex
negate(Apcomplex z)
Deprecated.UseApcomplex.negate()
.static Apfloat
norm(Apcomplex z)
Norm.(package private) static Apcomplex
pochhammer(Apcomplex z, long n)
static Apcomplex
pochhammer(Apcomplex z, Apcomplex n)
Pochhammer symbol.static Apcomplex
polygamma(long n, Apcomplex z)
Polygamma function.static Apcomplex
polylog(Apcomplex ν, Apcomplex z)
Polylogarithm.static Apcomplex
pow(Apcomplex z, long n)
Integer power.static Apcomplex
pow(Apcomplex z, Apcomplex w)
Arbitrary power.private static Apcomplex
powAbs(Apcomplex z, long n)
static Apcomplex
product(Apcomplex... z)
Product of numbers.private static Apcomplex
rawLog(Apcomplex z)
private static Apcomplex
rightSqrt(Apcomplex z, Apcomplex reference)
static Apcomplex
root(Apcomplex z, long n)
Positive integer root.static Apcomplex
root(Apcomplex z, long n, long k)
Positive integer root.static Apcomplex
scale(Apcomplex z, long scale)
Multiply by a power of the radix.static Apcomplex
sin(Apcomplex z)
Sine.static Apcomplex
sinc(Apcomplex z)
Sinc.static Apcomplex
sinh(Apcomplex z)
Hyperbolic sine.static Apcomplex
sinhIntegral(Apcomplex z)
Hyperbolic sine integral.static Apcomplex
sinIntegral(Apcomplex z)
Sine integral.static Apcomplex
sphericalHarmonicY(Apcomplex λ, Apcomplex μ, Apcomplex ϑ, Apcomplex ϕ)
Spherical harmonic function.private static Apcomplex
sphericalHarmonicY(Apint n, Apint m, Apcomplex ϑ, Apcomplex ϕ)
static Apcomplex
sqrt(Apcomplex z)
Square root.static Apcomplex
sum(Apcomplex... z)
Sum of numbers.static Apcomplex
tan(Apcomplex z)
Tangent.(package private) static Apcomplex
tan(Apcomplex z, boolean negate)
(package private) static Apcomplex
tanFixedPrecision(Apcomplex z)
static Apcomplex
tanh(Apcomplex z)
Hyperbolic tangent.private static Apcomplex
tanh(Apcomplex z, boolean negate)
(package private) static Apcomplex
tanhFixedPrecision(Apcomplex z)
static Apfloat
ulp(Apcomplex z)
Returns the unit in the last place of the argument, considering the scale and precision.static Apcomplex
w(Apcomplex z)
Lambert W function.static Apcomplex
w(Apcomplex z, long k)
Lambert W function for the specified branch.static Apcomplex
zeta(Apcomplex s)
Riemann zeta function.static Apcomplex
zeta(Apcomplex s, Apcomplex a)
Hurwitz zeta function.
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-
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Method Detail
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negate
@Deprecated public static Apcomplex negate(Apcomplex z) throws ApfloatRuntimeException
Deprecated.UseApcomplex.negate()
.Negative value.- Parameters:
z
- The argument.- Returns:
-z
.- Throws:
ApfloatRuntimeException
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abs
public static Apfloat abs(Apcomplex z) throws ApfloatRuntimeException
Absolute value.- Parameters:
z
- The argument.- Returns:
sqrt(x2 + y2)
, wherez = x + i y
.- Throws:
ApfloatRuntimeException
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norm
public static Apfloat norm(Apcomplex z) throws ApfloatRuntimeException
Norm. Square of the magnitude.- Parameters:
z
- The argument.- Returns:
x2 + y2
, wherez = x + i y
.- Throws:
ApfloatRuntimeException
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arg
public static Apfloat arg(Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Angle of the complex vector in the complex plane.- Parameters:
z
- The argument.- Returns:
arctan(y / x)
from the appropriate branch, wherez = x + i y
.- Throws:
java.lang.ArithmeticException
- Ifz
is zero.ApfloatRuntimeException
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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
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pow
public static Apcomplex pow(Apcomplex z, long n) throws java.lang.ArithmeticException, ApfloatRuntimeException
Integer power.- Parameters:
z
- Base of the power operator.n
- Exponent of the power operator.- Returns:
z
to then
:th power, that iszn
.- Throws:
java.lang.ArithmeticException
- If bothz
andn
are zero.ApfloatRuntimeException
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powAbs
private static Apcomplex powAbs(Apcomplex z, long n) throws java.lang.ArithmeticException, ApfloatRuntimeException
- Throws:
java.lang.ArithmeticException
ApfloatRuntimeException
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sqrt
public static Apcomplex sqrt(Apcomplex z) throws ApfloatRuntimeException
Square root.- Parameters:
z
- The argument.- Returns:
- Square root of
z
. - Throws:
ApfloatRuntimeException
-
cbrt
public static Apcomplex cbrt(Apcomplex z) throws ApfloatRuntimeException
Cube root.- Parameters:
z
- The argument.- Returns:
- Cube root of
z
. - Throws:
ApfloatRuntimeException
-
root
public static Apcomplex root(Apcomplex z, long n) throws java.lang.ArithmeticException, ApfloatRuntimeException
Positive integer root. The branch that has the smallest angle and same sign of imaginary part asz
is always chosen.- Parameters:
z
- The argument.n
- Which root to take.- Returns:
n
:th root ofz
, that isz1/n
.- Throws:
java.lang.ArithmeticException
- Ifn
is zero.ApfloatRuntimeException
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root
public static Apcomplex root(Apcomplex z, long n, long k) throws java.lang.ArithmeticException, ApfloatRuntimeException
Positive integer root. The specified branch counting from the smallest angle and same sign of imaginary part asz
is chosen.- Parameters:
z
- The argument.n
- Which root to take.k
- Which branch to take.- Returns:
n
:th root ofz
, that isz1/nei2πsk/n
wheres
is the signum of the imaginary part ofz
.- Throws:
java.lang.ArithmeticException
- Ifn
is zero.ApfloatRuntimeException
- Since:
- 1.5
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inverseRoot
public static Apcomplex inverseRoot(Apcomplex z, long n) throws java.lang.ArithmeticException, ApfloatRuntimeException
Inverse positive integer root. The branch that has the smallest angle and different sign of imaginary part thanz
is always chosen.- Parameters:
z
- The argument.n
- Which inverse root to take.- Returns:
- Inverse
n
:th root ofz
, that isz-1/n
. - Throws:
java.lang.ArithmeticException
- Ifz
orn
is zero.ApfloatRuntimeException
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inverseRoot
public static Apcomplex inverseRoot(Apcomplex z, long n, long k) throws java.lang.ArithmeticException, ApfloatRuntimeException
Inverse positive integer root. The specified branch counting from the smallest angle and different sign of imaginary part thanz
is chosen.- Parameters:
z
- The argument.n
- Which inverse root to take.k
- Which branch to take.- Returns:
- Inverse
n
:th root ofz
, that isz-1/ne-i2πk/n
. - Throws:
java.lang.ArithmeticException
- Ifz
orn
is zero.ApfloatRuntimeException
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inverseRootAbs
private static Apcomplex inverseRootAbs(Apcomplex z, long n, long k) throws java.lang.ArithmeticException, ApfloatRuntimeException
- Throws:
java.lang.ArithmeticException
ApfloatRuntimeException
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allRoots
public static Apcomplex[] allRoots(Apcomplex z, int n) throws java.lang.ArithmeticException, ApfloatRuntimeException
All values of the positive integer root.Returns all of the
n
values of the root, in the order of the angle, starting from the smallest angle and same sign of imaginary part asz
.- Parameters:
z
- The argument.n
- Which root to take.- Returns:
- All values of the
n
:th root ofz
, that isz1/n
, in the order of the angle. - Throws:
java.lang.ArithmeticException
- Ifn
is zero.ApfloatRuntimeException
- Since:
- 1.5
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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
andb
. - Throws:
ApfloatRuntimeException
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agm
static Apcomplex agm(Apcomplex a, Apcomplex b, java.util.function.Consumer<Apcomplex> consumer) throws ApfloatRuntimeException
- Throws:
ApfloatRuntimeException
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agmConsume
private static Apcomplex agmConsume(java.util.function.Consumer<Apcomplex> consumer, Apcomplex a, Apcomplex c2, long workingPrecision)
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log
public static Apcomplex log(Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Natural logarithm.The logarithm is calculated using the arithmetic-geometric mean. See the Borweins' book for the formula.
- Parameters:
z
- The argument.- Returns:
- Natural logarithm of
z
. - Throws:
java.lang.ArithmeticException
- Ifz
is zero.ApfloatRuntimeException
-
log
public static Apcomplex log(Apcomplex z, Apcomplex w) throws java.lang.ArithmeticException, ApfloatRuntimeException
Logarithm in arbitrary base.- Parameters:
z
- The argument.w
- The base.- Returns:
- Base-
w
logarithm ofz
. - Throws:
java.lang.ArithmeticException
- Ifz
orw
is zero.ApfloatRuntimeException
- Since:
- 1.6
-
rawLog
private static Apcomplex rawLog(Apcomplex z) throws ApfloatRuntimeException
- Throws:
ApfloatRuntimeException
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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 usinglog()
andexp()
.- Parameters:
z
- The base.w
- The exponent.- Returns:
zw
.- Throws:
java.lang.ArithmeticException
- If bothz
andw
are zero.ApfloatRuntimeException
-
acos
public static Apcomplex acos(Apcomplex z) throws ApfloatRuntimeException
Inverse cosine. Calculated usinglog()
.- Parameters:
z
- The argument.- Returns:
- Inverse cosine of
z
. - Throws:
ApfloatRuntimeException
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acosh
public static Apcomplex acosh(Apcomplex z) throws ApfloatRuntimeException
Inverse hyperbolic cosine. Calculated usinglog()
.- Parameters:
z
- The argument.- Returns:
- Inverse hyperbolic cosine of
z
. - Throws:
ApfloatRuntimeException
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asin
public static Apcomplex asin(Apcomplex z) throws ApfloatRuntimeException
Inverse sine. Calculated usinglog()
.- Parameters:
z
- The argument.- Returns:
- Inverse sine of
z
. - Throws:
ApfloatRuntimeException
-
asinh
public static Apcomplex asinh(Apcomplex z) throws ApfloatRuntimeException
Inverse hyperbolic sine. Calculated usinglog()
.- Parameters:
z
- The argument.- Returns:
- Inverse hyperbolic sine of
z
. - Throws:
ApfloatRuntimeException
-
atan
public static Apcomplex atan(Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Inverse tangent. Calculated usinglog()
.- Parameters:
z
- The argument.- Returns:
- Inverse tangent of
z
. - Throws:
java.lang.ArithmeticException
- Ifz == i
.ApfloatRuntimeException
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atanh
public static Apcomplex atanh(Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Inverse hyperbolic tangent. Calculated usinglog()
.- Parameters:
z
- The argument.- Returns:
- Inverse hyperbolic tangent of
z
. - Throws:
java.lang.ArithmeticException
- Ifz
is 1 or -1.ApfloatRuntimeException
-
cos
public static Apcomplex cos(Apcomplex z) throws ApfloatRuntimeException
Cosine. Calculated usingexp()
.- Parameters:
z
- The argument.- Returns:
- Cosine of
z
. - Throws:
ApfloatRuntimeException
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cosh
public static Apcomplex cosh(Apcomplex z) throws ApfloatRuntimeException
Hyperbolic cosine. Calculated usingexp()
.- Parameters:
z
- The argument.- Returns:
- Hyperbolic cosine of
z
. - Throws:
ApfloatRuntimeException
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sin
public static Apcomplex sin(Apcomplex z) throws ApfloatRuntimeException
Sine. Calculated usingexp()
.- Parameters:
z
- The argument.- Returns:
- Sine of
z
. - Throws:
ApfloatRuntimeException
-
sinh
public static Apcomplex sinh(Apcomplex z) throws ApfloatRuntimeException
Hyperbolic sine. Calculated usingexp()
.- Parameters:
z
- The argument.- Returns:
- Hyperbolic sine of
z
. - Throws:
ApfloatRuntimeException
-
tan
public static Apcomplex tan(Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Tangent. Calculated usingexp()
.- Parameters:
z
- The argument.- Returns:
- Tangent of
z
. - Throws:
java.lang.ArithmeticException
- Ifz
is π/2 + n π where n is an integer.ApfloatRuntimeException
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tanFixedPrecision
static Apcomplex tanFixedPrecision(Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
- Throws:
java.lang.ArithmeticException
ApfloatRuntimeException
-
tan
static Apcomplex tan(Apcomplex z, boolean negate) throws java.lang.ArithmeticException, ApfloatRuntimeException
- Throws:
java.lang.ArithmeticException
ApfloatRuntimeException
-
tanh
public static Apcomplex tanh(Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Hyperbolic tangent. Calculated usingexp()
.- Parameters:
z
- The argument.- Returns:
- Hyperbolic tangent of
z
. - Throws:
java.lang.ArithmeticException
- Ifz
is i (π/2 + n π) where n is an integer.ApfloatRuntimeException
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tanhFixedPrecision
static Apcomplex tanhFixedPrecision(Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
- Throws:
java.lang.ArithmeticException
ApfloatRuntimeException
-
tanh
private static Apcomplex tanh(Apcomplex z, boolean negate) throws java.lang.ArithmeticException, ApfloatRuntimeException
- Throws:
java.lang.ArithmeticException
ApfloatRuntimeException
-
cot
static Apcomplex cot(Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
- Throws:
java.lang.ArithmeticException
ApfloatRuntimeException
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sinc
public static Apcomplex sinc(Apcomplex z) throws ApfloatRuntimeException
Sinc.- Parameters:
z
- The argument.- Returns:
- sinc(z)
- Throws:
ApfloatRuntimeException
- Since:
- 1.14.0
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w
public static Apcomplex w(Apcomplex z) throws ApfloatRuntimeException
Lambert W function. The W function gives the solution to the equationW 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 java.lang.ArithmeticException, ApfloatRuntimeException
Lambert W function for the specified branch.- Parameters:
z
- The argument.k
- The branch.- Returns:
Wk(z)
.- Throws:
java.lang.ArithmeticException
- Ifz
is zero andk
is not zero.ApfloatRuntimeException
- Since:
- 1.8.0
- See Also:
w(Apcomplex)
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product
public static Apcomplex product(Apcomplex... z) throws ApfloatRuntimeException
Product of numbers. The precision used in the multiplications is only what is needed for the end result. This method may perform significantly better than simply multiplying the numbers sequentially.If there are no arguments, the return value is
1
.- Parameters:
z
- The argument(s).- Returns:
- The product of the given numbers.
- Throws:
ApfloatRuntimeException
- Since:
- 1.3
-
sum
public static Apcomplex sum(Apcomplex... z) throws ApfloatRuntimeException
Sum of numbers. The precision used in the additions is only what is needed for the end result. This method may perform significantly better than simply adding the numbers sequentially.If there are no arguments, the return value is
0
.- Parameters:
z
- The argument(s).- Returns:
- The sum of the given numbers.
- Throws:
ApfloatRuntimeException
- Since:
- 1.3
-
gamma
public static Apcomplex gamma(Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Gamma function.- Parameters:
z
- The argument.- Returns:
Γ(z)
- Throws:
java.lang.ArithmeticException
- Ifz
is a nonpositive integer.ApfloatRuntimeException
- Since:
- 1.9.0
-
gamma
public static Apcomplex gamma(Apcomplex a, Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Incomplete gamma function.- Parameters:
a
- The first argument.z
- The second argument.- Returns:
Γ(a, z)
- Throws:
java.lang.ArithmeticException
- If the real part ofa
is nonpositive andz
is zero.ApfloatRuntimeException
- Since:
- 1.10.0
-
gamma
public static Apcomplex gamma(Apcomplex a, Apcomplex z0, Apcomplex z1) throws java.lang.ArithmeticException, ApfloatRuntimeException
Generalized incomplete gamma function.This function is defined as:
Γ(a, z0, z1) = Γ(a, z0) - Γ(a, z1)
The lower gamma function can be calculated with:
γ(a, z) = Γ(a, 0, z)
- Parameters:
a
- The first argument.z0
- The second argument.z1
- The third argument.- Returns:
Γ(a, z0, z1)
- Throws:
java.lang.ArithmeticException
- If the real part ofa
is nonpositive and eitherz0
orz1
is zero. For the lower gamma function ifa
is a nonpositive integer.ApfloatRuntimeException
- Since:
- 1.10.0
-
logGamma
public static Apcomplex logGamma(Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Logarithm of the gamma function. Note that this function has a different branch structure thanlog(gamma(z))
.- Parameters:
z
- The argument.- Returns:
logΓ(z)
- Throws:
java.lang.ArithmeticException
- Ifz
is a nonpositive integer.ApfloatRuntimeException
- Since:
- 1.11.0
-
digamma
public static Apcomplex digamma(Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Digamma function.- Parameters:
z
- The argument.- Returns:
ψ(z)
- Throws:
java.lang.ArithmeticException
- Ifz
is a nonpositive integer.ApfloatRuntimeException
- Since:
- 1.11.0
-
polygamma
public static Apcomplex polygamma(long n, Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Polygamma function.- Parameters:
n
- The order.z
- The argument.- Returns:
ψ(n)(z)
- Throws:
java.lang.ArithmeticException
- Ifn
is negative orz
is a nonpositive integer.ApfloatRuntimeException
- Since:
- 1.13.0
-
beta
public static Apcomplex beta(Apcomplex a, Apcomplex b) throws java.lang.ArithmeticException, ApfloatRuntimeException
Beta function.- Parameters:
a
- The first argument.b
- The second argument.- Returns:
- B(a, b)
- Throws:
java.lang.ArithmeticException
- Ifa
orb
is a nonpositive integer buta + b
is not. Also if botha
andb
are nonpositive integers.ApfloatRuntimeException
- Since:
- 1.13.0
-
beta
public static Apcomplex beta(Apcomplex z, Apcomplex a, Apcomplex b) throws java.lang.ArithmeticException, ApfloatRuntimeException
Incomplete beta function.- Parameters:
z
- The first argument.a
- The second argument.b
- The third argument.- Returns:
- Bz(a, b)
- Throws:
java.lang.ArithmeticException
- Ifa
is a nonpositive integer orz
is zero anda
has nonpositive real part.ApfloatRuntimeException
- Since:
- 1.13.0
-
beta
public static Apcomplex beta(Apcomplex z1, Apcomplex z2, Apcomplex a, Apcomplex b) throws java.lang.ArithmeticException, ApfloatRuntimeException
Generalized incomplete beta function.- Parameters:
z1
- The first argument.z2
- The second argument.a
- The third argument.b
- The fourth argument.- Returns:
- B(z1, z2)(a, b)
- Throws:
java.lang.ArithmeticException
- Ifa
is a nonpositive integer orz1
orz2
is zero anda
has nonpositive real part.ApfloatRuntimeException
- Since:
- 1.13.0
-
pochhammer
public static Apcomplex pochhammer(Apcomplex z, Apcomplex n) throws java.lang.ArithmeticException, ApfloatRuntimeException
Pochhammer symbol.- Parameters:
z
- The first argument.n
- The second argument.- Returns:
(z)n
- Throws:
java.lang.ArithmeticException
- Ifz + n
is a nonpositive integer butz
is not.ApfloatRuntimeException
- Since:
- 1.13.0
-
binomial
public static Apcomplex binomial(Apcomplex n, Apcomplex k) throws java.lang.ArithmeticException, ApfloatRuntimeException
Binomial coefficient. Calculated using thegamma(Apcomplex)
function.- Parameters:
n
- The first argument.k
- The second argument.- Returns:
- Throws:
java.lang.ArithmeticException
- Ifn
is a negative integer andk
is noninteger.ApfloatRuntimeException
- Since:
- 1.11.0
-
zeta
public static Apcomplex zeta(Apcomplex s) throws java.lang.ArithmeticException, ApfloatRuntimeException
Riemann zeta function.- Parameters:
s
- The argument.- Returns:
ζ(s)
- Throws:
java.lang.ArithmeticException
- Ifs
is1
.ApfloatRuntimeException
- Since:
- 1.11.0
-
zeta
public static Apcomplex zeta(Apcomplex s, Apcomplex a) throws java.lang.ArithmeticException, ApfloatRuntimeException
Hurwitz zeta function.- Parameters:
s
- The first argument.a
- The second argument.- Returns:
ζ(s, a)
- Throws:
java.lang.ArithmeticException
- Ifs
is1
or ifa
is a nonpositive integer.ApfloatRuntimeException
- Since:
- 1.11.0
-
hypergeometric0F1
public static Apcomplex hypergeometric0F1(Apcomplex a, Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Confluent hypergeometric function 0F1.- Parameters:
a
- The first argument.z
- The second argument.- Returns:
- 0F1(; a; z)
- Throws:
java.lang.ArithmeticException
- If the function value is not finite.ApfloatRuntimeException
- Since:
- 1.11.0
-
hypergeometric0F1Regularized
public static Apcomplex hypergeometric0F1Regularized(Apcomplex a, Apcomplex z) throws ApfloatRuntimeException
Regularized confluent hypergeometric function 0F̃1.- Parameters:
a
- The first argument.z
- The second argument.- Returns:
- 0F̃1(; a; z)
- Throws:
ApfloatRuntimeException
- Since:
- 1.13.0
-
hypergeometric1F1
public static Apcomplex hypergeometric1F1(Apcomplex a, Apcomplex b, Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Kummer confluent hypergeometric function 1F1. Also known as the confluent hypergeometric function of the first kind.- Parameters:
a
- The first argument.b
- The second argument.z
- The third argument.- Returns:
- 1F1(a; b; z)
- Throws:
java.lang.ArithmeticException
- If the function value is not finite.ApfloatRuntimeException
- Since:
- 1.11.0
-
hypergeometric1F1Regularized
public static Apcomplex hypergeometric1F1Regularized(Apcomplex a, Apcomplex b, Apcomplex z) throws ApfloatRuntimeException
Regularized Kummer confluent hypergeometric function 1F̃1. Also known as the regularized confluent hypergeometric function of the first kind.- Parameters:
a
- The first argument.b
- The second argument.z
- The third argument.- Returns:
- 1F̃1(a; b; z)
- Throws:
ApfloatRuntimeException
- Since:
- 1.13.0
-
hypergeometric2F1
public static Apcomplex hypergeometric2F1(Apcomplex a, Apcomplex b, Apcomplex c, Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Hypergeometric function 2F1. Also known as the Gaussian or ordinary hypergeometric function.- Parameters:
a
- The first argument.b
- The second argument.c
- The third argument.z
- The fourth argument.- Returns:
- 2F1(a, b; c; z)
- Throws:
java.lang.ArithmeticException
- If the function value is not finite.ApfloatRuntimeException
- Since:
- 1.11.0
-
hypergeometric2F1Regularized
public static Apcomplex hypergeometric2F1Regularized(Apcomplex a, Apcomplex b, Apcomplex c, Apcomplex z) throws ApfloatRuntimeException
Regularized hypergeometric function 2F̃1. Also known as the regularized Gaussian or ordinary hypergeometric function.- Parameters:
a
- The first argument.b
- The second argument.c
- The third argument.z
- The fourth argument.- Returns:
- 2F̃1(a, b; c; z)
- Throws:
ApfloatRuntimeException
- Since:
- 1.13.0
-
hypergeometricU
public static Apcomplex hypergeometricU(Apcomplex a, Apcomplex b, Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Tricomi's confluent hypergeometric function U. Also known as the confluent hypergeometric function of the second kind.- Parameters:
a
- The first argument.b
- The second argument.z
- The third argument.- Returns:
- U(a, b, z)
- Throws:
java.lang.ArithmeticException
- If the result is not finite.ApfloatRuntimeException
- Since:
- 1.13.0
-
erf
public static Apcomplex erf(Apcomplex z) throws ApfloatRuntimeException
Error function.- Parameters:
z
- The argument.- Returns:
- erf(z)
- Throws:
ApfloatRuntimeException
- Since:
- 1.13.0
-
erfFixedPrecision
static Apcomplex erfFixedPrecision(Apcomplex z) throws ApfloatRuntimeException
- Throws:
ApfloatRuntimeException
-
erfc
public static Apcomplex erfc(Apcomplex z) throws ApfloatRuntimeException
Complementary error function.- Parameters:
z
- The argument.- Returns:
- erfc(z)
- Throws:
ApfloatRuntimeException
- Since:
- 1.13.0
-
erfcFixedPrecision
static Apcomplex erfcFixedPrecision(Apcomplex z) throws ApfloatRuntimeException
- Throws:
ApfloatRuntimeException
-
erfi
public static Apcomplex erfi(Apcomplex z) throws ApfloatRuntimeException
Imaginary error function.- Parameters:
z
- The argument.- Returns:
- erfi(z)
- Throws:
ApfloatRuntimeException
- Since:
- 1.13.0
-
erfiFixedPrecision
static Apcomplex erfiFixedPrecision(Apcomplex z) throws ApfloatRuntimeException
- Throws:
ApfloatRuntimeException
-
fresnelS
public static Apcomplex fresnelS(Apcomplex z) throws ApfloatRuntimeException
Fresnel integral S.- Parameters:
z
- The argument.- Returns:
- S(z)
- Throws:
ApfloatRuntimeException
- Since:
- 1.13.0
-
fresnelC
public static Apcomplex fresnelC(Apcomplex z) throws ApfloatRuntimeException
Fresnel integral C.- Parameters:
z
- The argument.- Returns:
- C(z)
- Throws:
ApfloatRuntimeException
- Since:
- 1.13.0
-
fresnelTerm
private static Apcomplex fresnelTerm(Apint one, Apfloat half, Apfloat invSqrtPi, Apcomplex iz2, Apcomplex iHalfPiZ2) throws ApfloatRuntimeException
- Throws:
ApfloatRuntimeException
-
expIntegralE
public static Apcomplex expIntegralE(Apcomplex ν, Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Exponential integral E.- Parameters:
\u03bd
- The first argument.z
- The second argument.- Returns:
- Eν(z)
- Throws:
java.lang.ArithmeticException
- If real part ofν
is ≤ 1 andz
is zero.ApfloatRuntimeException
- Since:
- 1.13.0
-
expIntegralEi
public static Apcomplex expIntegralEi(Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Exponential integral Ei.- Parameters:
z
- The argument.- Returns:
- Ei(z)
- Throws:
java.lang.ArithmeticException
- Ifz
is zero.ApfloatRuntimeException
- Since:
- 1.13.0
-
logIntegral
public static Apcomplex logIntegral(Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Logarithmic integral.- Parameters:
z
- The argument.- Returns:
- li(z)
- Throws:
java.lang.ArithmeticException
- Ifz
is zero.ApfloatRuntimeException
- Since:
- 1.13.0
-
sinIntegral
public static Apcomplex sinIntegral(Apcomplex z) throws ApfloatRuntimeException
Sine integral.- Parameters:
z
- The argument.- Returns:
- Si(z)
- Throws:
ApfloatRuntimeException
- Since:
- 1.13.0
-
cosIntegral
public static Apcomplex cosIntegral(Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Cosine integral.- Parameters:
z
- The argument.- Returns:
- Ci(z)
- Throws:
java.lang.ArithmeticException
- Ifz
is zero.ApfloatRuntimeException
- Since:
- 1.13.0
-
sinhIntegral
public static Apcomplex sinhIntegral(Apcomplex z) throws ApfloatRuntimeException
Hyperbolic sine integral.- Parameters:
z
- The argument.- Returns:
- Shi(z)
- Throws:
ApfloatRuntimeException
- Since:
- 1.13.0
-
coshIntegral
public static Apcomplex coshIntegral(Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Hyperbolic cosine integral.- Parameters:
z
- The argument.- Returns:
- Chi(z)
- Throws:
java.lang.ArithmeticException
- Ifz
is zero.ApfloatRuntimeException
- Since:
- 1.13.0
-
airyAi
public static Apcomplex airyAi(Apcomplex z) throws ApfloatRuntimeException
Airy function Ai.- Parameters:
z
- The argument.- Returns:
- Ai(z)
- Throws:
InfiniteExpansionException
- Ifz
is zero.ApfloatRuntimeException
- Since:
- 1.13.0
-
airyAi
static Apcomplex airyAi(Apcomplex z0, long targetPrecision) throws ApfloatRuntimeException
- Throws:
ApfloatRuntimeException
-
airyAiPrime
public static Apcomplex airyAiPrime(Apcomplex z) throws ApfloatRuntimeException
Derivative of the Airy function Ai.- Parameters:
z
- The argument.- Returns:
- Ai′(z)
- Throws:
InfiniteExpansionException
- Ifz
is zero.ApfloatRuntimeException
- Since:
- 1.13.0
-
airyAiPrime
static Apcomplex airyAiPrime(Apcomplex z0, long targetPrecision) throws ApfloatRuntimeException
- Throws:
ApfloatRuntimeException
-
airyBi
public static Apcomplex airyBi(Apcomplex z) throws ApfloatRuntimeException
Airy function Bi.- Parameters:
z
- The argument.- Returns:
- Bi(z)
- Throws:
InfiniteExpansionException
- Ifz
is zero.ApfloatRuntimeException
- Since:
- 1.13.0
-
airyBi
static Apcomplex airyBi(Apcomplex z0, long targetPrecision) throws ApfloatRuntimeException
- Throws:
ApfloatRuntimeException
-
airyBiPrime
public static Apcomplex airyBiPrime(Apcomplex z) throws ApfloatRuntimeException
Derivative of the Airy function Bi.- Parameters:
z
- The argument.- Returns:
- Bi′(z)
- Throws:
InfiniteExpansionException
- Ifz
is zero.ApfloatRuntimeException
- Since:
- 1.13.0
-
airyBiPrime
static Apcomplex airyBiPrime(Apcomplex z0, long targetPrecision) throws ApfloatRuntimeException
- Throws:
ApfloatRuntimeException
-
airy
private static Apcomplex airy(java.util.function.Function<java.lang.Long,Apcomplex> f, long targetPrecision, int radix)
-
besselJ
public static Apcomplex besselJ(Apcomplex ν, Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Bessel function of the first kind.- Parameters:
\u03bd
- The order.z
- The argument.- Returns:
- Jν(z)
- Throws:
java.lang.ArithmeticException
- If the real part ofν
is < 0 andν
is not an integer andz
is zero. Also if the real part ofν
is zero but the imaginary part is not, andz
is zero.ApfloatRuntimeException
- Since:
- 1.13.0
-
besselI
public static Apcomplex besselI(Apcomplex ν, Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Modified Bessel function of the first kind.- Parameters:
\u03bd
- The order.z
- The argument.- Returns:
- Iν(z)
- Throws:
java.lang.ArithmeticException
- If the real part ofν
is < 0 andν
is not an integer andz
is zero. Also if the real part ofν
is zero but the imaginary part is not, andz
is zero.ApfloatRuntimeException
- Since:
- 1.13.0
-
besselY
public static Apcomplex besselY(Apcomplex ν, Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Bessel function of the second kind.- Parameters:
\u03bd
- The order.z
- The argument.- Returns:
- Yν(z)
- Throws:
java.lang.ArithmeticException
- Ifz
is zero.ApfloatRuntimeException
- Since:
- 1.13.0
-
besselK
public static Apcomplex besselK(Apcomplex ν, Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Modified Bessel function of the second kind.- Parameters:
\u03bd
- The order.z
- The argument.- Returns:
- Kν(z)
- Throws:
java.lang.ArithmeticException
- Ifz
is zero.ApfloatRuntimeException
- Since:
- 1.13.0
-
ellipticK
public static Apcomplex ellipticK(Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Complete elliptic integral of the first kind.Note that this function uses the definition:
- Parameters:
z
- The argument.- Returns:
- K(z)
- Throws:
InfiniteExpansionException
- Ifz
is zero.java.lang.ArithmeticException
- Ifz
is one.ApfloatRuntimeException
- Since:
- 1.13.0
-
ellipticK
static Apcomplex ellipticK(Apcomplex z, long precision) throws java.lang.ArithmeticException, ApfloatRuntimeException
- Throws:
java.lang.ArithmeticException
ApfloatRuntimeException
-
ellipticK
static Apcomplex ellipticK(Apcomplex z, long precision, java.util.function.Consumer<Apcomplex> consumer) throws java.lang.ArithmeticException, ApfloatRuntimeException
- Throws:
java.lang.ArithmeticException
ApfloatRuntimeException
-
ellipticE
public static Apcomplex ellipticE(Apcomplex z) throws ApfloatRuntimeException
Complete elliptic integral of the second kind.Note that this function uses the definition:
- Parameters:
z
- The argument.- Returns:
- E(z)
- Throws:
InfiniteExpansionException
- Ifz
is zero.ApfloatRuntimeException
- Since:
- 1.13.0
-
ellipticE
static Apcomplex ellipticE(Apcomplex z, long precision) throws ApfloatRuntimeException
- Throws:
ApfloatRuntimeException
-
hermiteH
public static Apcomplex hermiteH(Apcomplex ν, Apcomplex z) throws ApfloatRuntimeException
Hermite function. For integer values ofν
gives the Hermite polynomial.- Parameters:
\u03bd
- The first argument.z
- The second argument.- Returns:
- Hν(z)
- Throws:
ApfloatRuntimeException
- Since:
- 1.14.0
-
laguerreL
public static Apcomplex laguerreL(Apcomplex ν, Apcomplex z) throws ApfloatRuntimeException
Laguerre function. For integer values ofν
gives the Laguerre polynomial.- Parameters:
\u03bd
- The first argument.z
- The second argument.- Returns:
- Lν(z)
- Throws:
ApfloatRuntimeException
- Since:
- 1.14.0
-
laguerreL
public static Apcomplex laguerreL(Apcomplex ν, Apcomplex λ, Apcomplex z) throws ApfloatRuntimeException
Generalized Laguerre function. For integer values ofν
gives the generalized Laguerre polynomial.- Parameters:
\u03bd
- The first argument.\u03bb
- The second argument.z
- The third argument.- Returns:
- Lνλ(z)
- Throws:
ApfloatRuntimeException
- Since:
- 1.14.0
-
legendreP
public static Apcomplex legendreP(Apcomplex ν, Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Legendre function. For integer values ofν
gives the Legendre polynomial.- Parameters:
\u03bd
- The first argument.z
- The second argument.- Returns:
- Pν(z)
- Throws:
java.lang.ArithmeticException
- Ifν
is not an integer andz
is -1.ApfloatRuntimeException
- Since:
- 1.14.0
-
legendreP
public static Apcomplex legendreP(Apcomplex ν, Apcomplex μ, Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Associated Legendre function of the first kind. Gives Legendre functions of type 2.- Parameters:
\u03bd
- The first argument.\u03bc
- The second argument.z
- The third argument.- Returns:
- Pνμ(z)
- Throws:
java.lang.ArithmeticException
- Ifν
is not an integer andz
is -1.ApfloatRuntimeException
- Since:
- 1.14.0
-
legendreQ
public static Apcomplex legendreQ(Apcomplex ν, Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Legendre function of the second kind.- Parameters:
\u03bd
- The first argument.z
- The second argument.- Returns:
- Qν(z)
- Throws:
java.lang.ArithmeticException
- Ifz
is 1 or -1.ApfloatRuntimeException
- Since:
- 1.14.0
-
legendreQ
public static Apcomplex legendreQ(Apcomplex ν, Apcomplex μ, Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Associated Legendre function of the second kind. Gives Legendre functions of type 2.- Parameters:
\u03bd
- The first argument.\u03bc
- The second argument.z
- The third argument.- Returns:
- Qνμ(z)
- Throws:
java.lang.ArithmeticException
- Ifz
is 1 or -1.ApfloatRuntimeException
- Since:
- 1.14.0
-
sphericalHarmonicY
public static Apcomplex sphericalHarmonicY(Apcomplex λ, Apcomplex μ, Apcomplex ϑ, Apcomplex ϕ) throws java.lang.ArithmeticException, ApfloatRuntimeException
Spherical harmonic function.- Parameters:
\u03bb
- The first argument.\u03bc
- The second argument.\u03d1
- The third argument.\u03d5
- The fourth argument.- Returns:
- Yλμ(ϑ, φ)
- Throws:
java.lang.ArithmeticException
- Ifϑ
is π plus a multiple of 2 π and μ is not an integer and has a negative real part, or ifλ - μ
is a negative integer.ApfloatRuntimeException
- Since:
- 1.14.0
-
sphericalHarmonicY
private static Apcomplex sphericalHarmonicY(Apint n, Apint m, Apcomplex ϑ, Apcomplex ϕ) throws java.lang.ArithmeticException, ApfloatRuntimeException
- Throws:
java.lang.ArithmeticException
ApfloatRuntimeException
-
chebyshevT
public static Apcomplex chebyshevT(Apcomplex ν, Apcomplex z) throws ApfloatRuntimeException
Chebyshev function of the first kind. For integer values ofν
gives the Chebyshev polynomial of the first kind.- Parameters:
\u03bd
- The first argument.z
- The second argument.- Returns:
- Tν(z)
- Throws:
ApfloatRuntimeException
- Since:
- 1.14.0
-
chebyshevU
public static Apcomplex chebyshevU(Apcomplex ν, Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Chebyshev function of the second kind. For integer values ofν
gives the Chebyshev polynomial of the second kind.- Parameters:
\u03bd
- The first argument.z
- The second argument.- Returns:
- Uν(z)
- Throws:
java.lang.ArithmeticException
- Ifz
is -1 andν
is not an integer.ApfloatRuntimeException
- Since:
- 1.14.0
-
gegenbauerC
public static Apcomplex gegenbauerC(Apcomplex ν, Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Renormalized Gegenbauer function.- Parameters:
\u03bd
- The first argument.z
- The second argument.- Returns:
- Cν(0)(z)
- Throws:
java.lang.ArithmeticException
- Ifν
is zero.ApfloatRuntimeException
- Since:
- 1.14.0
-
gegenbauerC
public static Apcomplex gegenbauerC(Apcomplex ν, Apcomplex λ, Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Gegenbauer function. For nonnegative integer values ofν
gives the Gegenbauer polynomial.- Parameters:
\u03bd
- The first argument.\u03bb
- The second argument.z
- The third argument.- Returns:
- Cνλ(z)
- Throws:
java.lang.ArithmeticException
- Ifz
is -1 and real part ofλ
is > 1/2. Also ifz
is -1 andλ
is 1/2 andν
is not an integer.ApfloatRuntimeException
- Since:
- 1.14.0
-
gegenbauerC
private static Apcomplex gegenbauerC(long n, Apcomplex λ, Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
- Throws:
java.lang.ArithmeticException
ApfloatRuntimeException
-
jacobiP
public static Apcomplex jacobiP(Apcomplex ν, Apcomplex a, Apcomplex b, Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Jacobi function. For nonnegative integer values ofν
gives the Jacobi polynomial.- Parameters:
\u03bd
- The first argument.a
- The second argument.b
- The third argument.z
- The fourth argument.- Returns:
- Pν(a,b)(z)
- Throws:
java.lang.ArithmeticException
- Ifz
is -1 and real part ofb
is > 0 andν
is not a positive integer. Also ifν + a
is a negative integer andν
is not an integer.ApfloatRuntimeException
- Since:
- 1.14.0
-
jacobiP
private static Apcomplex jacobiP(long n, Apcomplex a, Apcomplex b, Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
- Throws:
java.lang.ArithmeticException
ApfloatRuntimeException
-
fibonacci
public static Apcomplex fibonacci(Apcomplex ν, Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Fibonacci function. For nonnegative integer values ofν
gives the Fibonacci polynomial.- Parameters:
\u03bd
- The first argument.z
- The second argument.- Returns:
- Fν(z)
- Throws:
java.lang.ArithmeticException
- Ifz
is -1 andν
is not an integer.ApfloatRuntimeException
- Since:
- 1.14.0
-
eulerE
public static Apcomplex eulerE(long n, Apcomplex z) throws java.lang.IllegalArgumentException, ApfloatRuntimeException
Euler polynomial.- Parameters:
n
- The first argument.z
- The second argument.- Returns:
- En(z)
- Throws:
java.lang.IllegalArgumentException
- Ifn
< 0.ApfloatRuntimeException
- Since:
- 1.14.0
-
eulerE
static Apcomplex eulerE(long n, Apcomplex z, long precision) throws java.lang.IllegalArgumentException, ApfloatRuntimeException
- Throws:
java.lang.IllegalArgumentException
ApfloatRuntimeException
-
bernoulliB
public static Apcomplex bernoulliB(long n, Apcomplex z) throws java.lang.IllegalArgumentException, ApfloatRuntimeException
Bernoulli polynomial.- Parameters:
n
- The first argument.z
- The second argument.- Returns:
- Bn(z)
- Throws:
java.lang.IllegalArgumentException
- Ifn
< 0.ApfloatRuntimeException
- Since:
- 1.14.0
-
bernoulliB
static Apcomplex bernoulliB(long n, Apcomplex z, long precision) throws java.lang.IllegalArgumentException, ApfloatRuntimeException
- Throws:
java.lang.IllegalArgumentException
ApfloatRuntimeException
-
harmonicNumber
public static Apcomplex harmonicNumber(Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Harmonic number.- Parameters:
z
- The argument.- Returns:
- Hz
- Throws:
java.lang.ArithmeticException
- Ifz
is a negative integer.ApfloatRuntimeException
- Since:
- 1.14.0
-
harmonicNumber
public static Apcomplex harmonicNumber(Apcomplex z, Apcomplex r) throws java.lang.ArithmeticException, ApfloatRuntimeException
Generalized harmonic number.- Parameters:
z
- The first argument.r
- The second argument.- Returns:
- Hz(r)
- Throws:
java.lang.ArithmeticException
- Ifz
is a negative integer, unlessr
has a negative real part or is zero.ApfloatRuntimeException
- Since:
- 1.14.0
-
polylog
public static Apcomplex polylog(Apcomplex ν, Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Polylogarithm.- Parameters:
\u03bd
- The first argument.z
- The second argument.- Returns:
- Liν(z)
- Throws:
java.lang.ArithmeticException
- If the real part ofν
is ≤ 1 andz
is 1.ApfloatRuntimeException
- Since:
- 1.14.0
-
logisticSigmoid
public static Apcomplex logisticSigmoid(Apcomplex z) throws java.lang.ArithmeticException, ApfloatRuntimeException
Logistic sigmoid.- Parameters:
z
- The argument.- Returns:
- σ(z)
- Throws:
java.lang.ArithmeticException
- Ifz
is an odd integer multiple of π i.ApfloatRuntimeException
- Since:
- 1.14.0
-
ulp
public static Apfloat ulp(Apcomplex z)
Returns the unit in the last place of the argument, considering the scale and precision. This is maximum of the ulps of the real and imaginary part of the argument. If the precision of the argument is infinite, zero is returned.- Parameters:
z
- The argument.- Returns:
- The ulp of the argument.
- Since:
- 1.10.0
-
lastIterationExtendPrecision
private static Apcomplex lastIterationExtendPrecision(int iterations, int precisingIteration, Apcomplex z)
-
isNonPositiveInteger
static boolean isNonPositiveInteger(Apcomplex z)
-
-