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The Apache Commons Math project is a library of lightweight, self-contained mathematics and statistics components addressing the most common practical problems not immediately available in the Java programming language or commons-lang.
/*
* Licensed to the Apache Software Foundation (ASF) under one or more
* contributor license agreements. See the NOTICE file distributed with
* this work for additional information regarding copyright ownership.
* The ASF licenses this file to You under the Apache License, Version 2.0
* (the "License"); you may not use this file except in compliance with
* the License. You may obtain a copy of the License at
*
* http://www.apache.org/licenses/LICENSE-2.0
*
* Unless required by applicable law or agreed to in writing, software
* distributed under the License is distributed on an "AS IS" BASIS,
* WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
* See the License for the specific language governing permissions and
* limitations under the License.
*/
package org.apache.commons.math3.special;
import org.apache.commons.math3.exception.MaxCountExceededException;
import org.apache.commons.math3.exception.NumberIsTooLargeException;
import org.apache.commons.math3.exception.NumberIsTooSmallException;
import org.apache.commons.math3.util.ContinuedFraction;
import org.apache.commons.math3.util.FastMath;
/**
*
* This is a utility class that provides computation methods related to the
* Γ (Gamma) family of functions.
*
*
* Implementation of {@link #invGamma1pm1(double)} and
* {@link #logGamma1p(double)} is based on the algorithms described in
*
* - Didonato and Morris
* (1986), Computation of the Incomplete Gamma Function Ratios and
* their Inverse, TOMS 12(4), 377-393,
* - Didonato and Morris
* (1992), Algorithm 708: Significant Digit Computation of the
* Incomplete Beta Function Ratios, TOMS 18(3), 360-373,
*
* and implemented in the
* NSWC Library of Mathematical Functions,
* available
* here.
* This library is "approved for public release", and the
* Copyright guidance
* indicates that unless otherwise stated in the code, all FORTRAN functions in
* this library are license free. Since no such notice appears in the code these
* functions can safely be ported to Commons-Math.
*
*
*/
public class Gamma {
/**
* Euler-Mascheroni constant
* @since 2.0
*/
public static final double GAMMA = 0.577215664901532860606512090082;
/**
* The value of the {@code g} constant in the Lanczos approximation, see
* {@link #lanczos(double)}.
* @since 3.1
*/
public static final double LANCZOS_G = 607.0 / 128.0;
/** Maximum allowed numerical error. */
private static final double DEFAULT_EPSILON = 10e-15;
/** Lanczos coefficients */
private static final double[] LANCZOS = {
0.99999999999999709182,
57.156235665862923517,
-59.597960355475491248,
14.136097974741747174,
-0.49191381609762019978,
.33994649984811888699e-4,
.46523628927048575665e-4,
-.98374475304879564677e-4,
.15808870322491248884e-3,
-.21026444172410488319e-3,
.21743961811521264320e-3,
-.16431810653676389022e-3,
.84418223983852743293e-4,
-.26190838401581408670e-4,
.36899182659531622704e-5,
};
/** Avoid repeated computation of log of 2 PI in logGamma */
private static final double HALF_LOG_2_PI = 0.5 * FastMath.log(2.0 * FastMath.PI);
/** The constant value of √(2π). */
private static final double SQRT_TWO_PI = 2.506628274631000502;
// limits for switching algorithm in digamma
/** C limit. */
private static final double C_LIMIT = 49;
/** S limit. */
private static final double S_LIMIT = 1e-5;
/*
* Constants for the computation of double invGamma1pm1(double).
* Copied from DGAM1 in the NSWC library.
*/
/** The constant {@code A0} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_A0 = .611609510448141581788E-08;
/** The constant {@code A1} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_A1 = .624730830116465516210E-08;
/** The constant {@code B1} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_B1 = .203610414066806987300E+00;
/** The constant {@code B2} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_B2 = .266205348428949217746E-01;
/** The constant {@code B3} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_B3 = .493944979382446875238E-03;
/** The constant {@code B4} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_B4 = -.851419432440314906588E-05;
/** The constant {@code B5} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_B5 = -.643045481779353022248E-05;
/** The constant {@code B6} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_B6 = .992641840672773722196E-06;
/** The constant {@code B7} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_B7 = -.607761895722825260739E-07;
/** The constant {@code B8} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_B8 = .195755836614639731882E-09;
/** The constant {@code P0} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_P0 = .6116095104481415817861E-08;
/** The constant {@code P1} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_P1 = .6871674113067198736152E-08;
/** The constant {@code P2} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_P2 = .6820161668496170657918E-09;
/** The constant {@code P3} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_P3 = .4686843322948848031080E-10;
/** The constant {@code P4} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_P4 = .1572833027710446286995E-11;
/** The constant {@code P5} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_P5 = -.1249441572276366213222E-12;
/** The constant {@code P6} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_P6 = .4343529937408594255178E-14;
/** The constant {@code Q1} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_Q1 = .3056961078365221025009E+00;
/** The constant {@code Q2} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_Q2 = .5464213086042296536016E-01;
/** The constant {@code Q3} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_Q3 = .4956830093825887312020E-02;
/** The constant {@code Q4} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_Q4 = .2692369466186361192876E-03;
/** The constant {@code C} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_C = -.422784335098467139393487909917598E+00;
/** The constant {@code C0} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_C0 = .577215664901532860606512090082402E+00;
/** The constant {@code C1} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_C1 = -.655878071520253881077019515145390E+00;
/** The constant {@code C2} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_C2 = -.420026350340952355290039348754298E-01;
/** The constant {@code C3} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_C3 = .166538611382291489501700795102105E+00;
/** The constant {@code C4} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_C4 = -.421977345555443367482083012891874E-01;
/** The constant {@code C5} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_C5 = -.962197152787697356211492167234820E-02;
/** The constant {@code C6} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_C6 = .721894324666309954239501034044657E-02;
/** The constant {@code C7} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_C7 = -.116516759185906511211397108401839E-02;
/** The constant {@code C8} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_C8 = -.215241674114950972815729963053648E-03;
/** The constant {@code C9} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_C9 = .128050282388116186153198626328164E-03;
/** The constant {@code C10} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_C10 = -.201348547807882386556893914210218E-04;
/** The constant {@code C11} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_C11 = -.125049348214267065734535947383309E-05;
/** The constant {@code C12} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_C12 = .113302723198169588237412962033074E-05;
/** The constant {@code C13} defined in {@code DGAM1}. */
private static final double INV_GAMMA1P_M1_C13 = -.205633841697760710345015413002057E-06;
/**
* Default constructor. Prohibit instantiation.
*/
private Gamma() {}
/**
*
* Returns the value of log Γ(x) for x > 0.
*
*
* For x ≤ 8, the implementation is based on the double precision
* implementation in the NSWC Library of Mathematics Subroutines,
* {@code DGAMLN}. For x > 8, the implementation is based on
*
*
* - Gamma
* Function, equation (28).
* -
* Lanczos Approximation, equations (1) through (5).
* - Paul Godfrey, A note on
* the computation of the convergent Lanczos complex Gamma
* approximation
*
*
* @param x Argument.
* @return the value of {@code log(Gamma(x))}, {@code Double.NaN} if
* {@code x <= 0.0}.
*/
public static double logGamma(double x) {
double ret;
if (Double.isNaN(x) || (x <= 0.0)) {
ret = Double.NaN;
} else if (x < 0.5) {
return logGamma1p(x) - FastMath.log(x);
} else if (x <= 2.5) {
return logGamma1p((x - 0.5) - 0.5);
} else if (x <= 8.0) {
final int n = (int) FastMath.floor(x - 1.5);
double prod = 1.0;
for (int i = 1; i <= n; i++) {
prod *= x - i;
}
return logGamma1p(x - (n + 1)) + FastMath.log(prod);
} else {
double sum = lanczos(x);
double tmp = x + LANCZOS_G + .5;
ret = ((x + .5) * FastMath.log(tmp)) - tmp +
HALF_LOG_2_PI + FastMath.log(sum / x);
}
return ret;
}
/**
* Returns the regularized gamma function P(a, x).
*
* @param a Parameter.
* @param x Value.
* @return the regularized gamma function P(a, x).
* @throws MaxCountExceededException if the algorithm fails to converge.
*/
public static double regularizedGammaP(double a, double x) {
return regularizedGammaP(a, x, DEFAULT_EPSILON, Integer.MAX_VALUE);
}
/**
* Returns the regularized gamma function P(a, x).
*
* The implementation of this method is based on:
*
* -
*
* Regularized Gamma Function, equation (1)
*
* -
*
* Incomplete Gamma Function, equation (4).
*
* -
*
* Confluent Hypergeometric Function of the First Kind, equation (1).
*
*
*
* @param a the a parameter.
* @param x the value.
* @param epsilon When the absolute value of the nth item in the
* series is less than epsilon the approximation ceases to calculate
* further elements in the series.
* @param maxIterations Maximum number of "iterations" to complete.
* @return the regularized gamma function P(a, x)
* @throws MaxCountExceededException if the algorithm fails to converge.
*/
public static double regularizedGammaP(double a,
double x,
double epsilon,
int maxIterations) {
double ret;
if (Double.isNaN(a) || Double.isNaN(x) || (a <= 0.0) || (x < 0.0)) {
ret = Double.NaN;
} else if (x == 0.0) {
ret = 0.0;
} else if (x >= a + 1) {
// use regularizedGammaQ because it should converge faster in this
// case.
ret = 1.0 - regularizedGammaQ(a, x, epsilon, maxIterations);
} else {
// calculate series
double n = 0.0; // current element index
double an = 1.0 / a; // n-th element in the series
double sum = an; // partial sum
while (FastMath.abs(an/sum) > epsilon &&
n < maxIterations &&
sum < Double.POSITIVE_INFINITY) {
// compute next element in the series
n += 1.0;
an *= x / (a + n);
// update partial sum
sum += an;
}
if (n >= maxIterations) {
throw new MaxCountExceededException(maxIterations);
} else if (Double.isInfinite(sum)) {
ret = 1.0;
} else {
ret = FastMath.exp(-x + (a * FastMath.log(x)) - logGamma(a)) * sum;
}
}
return ret;
}
/**
* Returns the regularized gamma function Q(a, x) = 1 - P(a, x).
*
* @param a the a parameter.
* @param x the value.
* @return the regularized gamma function Q(a, x)
* @throws MaxCountExceededException if the algorithm fails to converge.
*/
public static double regularizedGammaQ(double a, double x) {
return regularizedGammaQ(a, x, DEFAULT_EPSILON, Integer.MAX_VALUE);
}
/**
* Returns the regularized gamma function Q(a, x) = 1 - P(a, x).
*
* The implementation of this method is based on:
*
* -
*
* Regularized Gamma Function, equation (1).
*
* -
*
* Regularized incomplete gamma function: Continued fraction representations
* (formula 06.08.10.0003)
*
*
*
* @param a the a parameter.
* @param x the value.
* @param epsilon When the absolute value of the nth item in the
* series is less than epsilon the approximation ceases to calculate
* further elements in the series.
* @param maxIterations Maximum number of "iterations" to complete.
* @return the regularized gamma function P(a, x)
* @throws MaxCountExceededException if the algorithm fails to converge.
*/
public static double regularizedGammaQ(final double a,
double x,
double epsilon,
int maxIterations) {
double ret;
if (Double.isNaN(a) || Double.isNaN(x) || (a <= 0.0) || (x < 0.0)) {
ret = Double.NaN;
} else if (x == 0.0) {
ret = 1.0;
} else if (x < a + 1.0) {
// use regularizedGammaP because it should converge faster in this
// case.
ret = 1.0 - regularizedGammaP(a, x, epsilon, maxIterations);
} else {
// create continued fraction
ContinuedFraction cf = new ContinuedFraction() {
@Override
protected double getA(int n, double x) {
return ((2.0 * n) + 1.0) - a + x;
}
@Override
protected double getB(int n, double x) {
return n * (a - n);
}
};
ret = 1.0 / cf.evaluate(x, epsilon, maxIterations);
ret = FastMath.exp(-x + (a * FastMath.log(x)) - logGamma(a)) * ret;
}
return ret;
}
/**
* Computes the digamma function of x.
*
* This is an independently written implementation of the algorithm described in
* Jose Bernardo, Algorithm AS 103: Psi (Digamma) Function, Applied Statistics, 1976.
*
* Some of the constants have been changed to increase accuracy at the moderate expense
* of run-time. The result should be accurate to within 10^-8 absolute tolerance for
* x >= 10^-5 and within 10^-8 relative tolerance for x > 0.
*
* Performance for large negative values of x will be quite expensive (proportional to
* |x|). Accuracy for negative values of x should be about 10^-8 absolute for results
* less than 10^5 and 10^-8 relative for results larger than that.
*
* @param x Argument.
* @return digamma(x) to within 10-8 relative or absolute error whichever is smaller.
* @see Digamma
* @see Bernardo's original article
* @since 2.0
*/
public static double digamma(double x) {
if (x > 0 && x <= S_LIMIT) {
// use method 5 from Bernardo AS103
// accurate to O(x)
return -GAMMA - 1 / x;
}
if (x >= C_LIMIT) {
// use method 4 (accurate to O(1/x^8)
double inv = 1 / (x * x);
// 1 1 1 1
// log(x) - --- - ------ + ------- - -------
// 2 x 12 x^2 120 x^4 252 x^6
return FastMath.log(x) - 0.5 / x - inv * ((1.0 / 12) + inv * (1.0 / 120 - inv / 252));
}
return digamma(x + 1) - 1 / x;
}
/**
* Computes the trigamma function of x.
* This function is derived by taking the derivative of the implementation
* of digamma.
*
* @param x Argument.
* @return trigamma(x) to within 10-8 relative or absolute error whichever is smaller
* @see Trigamma
* @see Gamma#digamma(double)
* @since 2.0
*/
public static double trigamma(double x) {
if (x > 0 && x <= S_LIMIT) {
return 1 / (x * x);
}
if (x >= C_LIMIT) {
double inv = 1 / (x * x);
// 1 1 1 1 1
// - + ---- + ---- - ----- + -----
// x 2 3 5 7
// 2 x 6 x 30 x 42 x
return 1 / x + inv / 2 + inv / x * (1.0 / 6 - inv * (1.0 / 30 + inv / 42));
}
return trigamma(x + 1) + 1 / (x * x);
}
/**
*
* Returns the Lanczos approximation used to compute the gamma function.
* The Lanczos approximation is related to the Gamma function by the
* following equation
*
* {@code gamma(x) = sqrt(2 * pi) / x * (x + g + 0.5) ^ (x + 0.5)
* * exp(-x - g - 0.5) * lanczos(x)},
*
* where {@code g} is the Lanczos constant.
*
*
* @param x Argument.
* @return The Lanczos approximation.
* @see Lanczos Approximation
* equations (1) through (5), and Paul Godfrey's
* Note on the computation
* of the convergent Lanczos complex Gamma approximation
* @since 3.1
*/
public static double lanczos(final double x) {
double sum = 0.0;
for (int i = LANCZOS.length - 1; i > 0; --i) {
sum += LANCZOS[i] / (x + i);
}
return sum + LANCZOS[0];
}
/**
* Returns the value of 1 / Γ(1 + x) - 1 for -0.5 ≤ x ≤
* 1.5. This implementation is based on the double precision
* implementation in the NSWC Library of Mathematics Subroutines,
* {@code DGAM1}.
*
* @param x Argument.
* @return The value of {@code 1.0 / Gamma(1.0 + x) - 1.0}.
* @throws NumberIsTooSmallException if {@code x < -0.5}
* @throws NumberIsTooLargeException if {@code x > 1.5}
* @since 3.1
*/
public static double invGamma1pm1(final double x) {
if (x < -0.5) {
throw new NumberIsTooSmallException(x, -0.5, true);
}
if (x > 1.5) {
throw new NumberIsTooLargeException(x, 1.5, true);
}
final double ret;
final double t = x <= 0.5 ? x : (x - 0.5) - 0.5;
if (t < 0.0) {
final double a = INV_GAMMA1P_M1_A0 + t * INV_GAMMA1P_M1_A1;
double b = INV_GAMMA1P_M1_B8;
b = INV_GAMMA1P_M1_B7 + t * b;
b = INV_GAMMA1P_M1_B6 + t * b;
b = INV_GAMMA1P_M1_B5 + t * b;
b = INV_GAMMA1P_M1_B4 + t * b;
b = INV_GAMMA1P_M1_B3 + t * b;
b = INV_GAMMA1P_M1_B2 + t * b;
b = INV_GAMMA1P_M1_B1 + t * b;
b = 1.0 + t * b;
double c = INV_GAMMA1P_M1_C13 + t * (a / b);
c = INV_GAMMA1P_M1_C12 + t * c;
c = INV_GAMMA1P_M1_C11 + t * c;
c = INV_GAMMA1P_M1_C10 + t * c;
c = INV_GAMMA1P_M1_C9 + t * c;
c = INV_GAMMA1P_M1_C8 + t * c;
c = INV_GAMMA1P_M1_C7 + t * c;
c = INV_GAMMA1P_M1_C6 + t * c;
c = INV_GAMMA1P_M1_C5 + t * c;
c = INV_GAMMA1P_M1_C4 + t * c;
c = INV_GAMMA1P_M1_C3 + t * c;
c = INV_GAMMA1P_M1_C2 + t * c;
c = INV_GAMMA1P_M1_C1 + t * c;
c = INV_GAMMA1P_M1_C + t * c;
if (x > 0.5) {
ret = t * c / x;
} else {
ret = x * ((c + 0.5) + 0.5);
}
} else {
double p = INV_GAMMA1P_M1_P6;
p = INV_GAMMA1P_M1_P5 + t * p;
p = INV_GAMMA1P_M1_P4 + t * p;
p = INV_GAMMA1P_M1_P3 + t * p;
p = INV_GAMMA1P_M1_P2 + t * p;
p = INV_GAMMA1P_M1_P1 + t * p;
p = INV_GAMMA1P_M1_P0 + t * p;
double q = INV_GAMMA1P_M1_Q4;
q = INV_GAMMA1P_M1_Q3 + t * q;
q = INV_GAMMA1P_M1_Q2 + t * q;
q = INV_GAMMA1P_M1_Q1 + t * q;
q = 1.0 + t * q;
double c = INV_GAMMA1P_M1_C13 + (p / q) * t;
c = INV_GAMMA1P_M1_C12 + t * c;
c = INV_GAMMA1P_M1_C11 + t * c;
c = INV_GAMMA1P_M1_C10 + t * c;
c = INV_GAMMA1P_M1_C9 + t * c;
c = INV_GAMMA1P_M1_C8 + t * c;
c = INV_GAMMA1P_M1_C7 + t * c;
c = INV_GAMMA1P_M1_C6 + t * c;
c = INV_GAMMA1P_M1_C5 + t * c;
c = INV_GAMMA1P_M1_C4 + t * c;
c = INV_GAMMA1P_M1_C3 + t * c;
c = INV_GAMMA1P_M1_C2 + t * c;
c = INV_GAMMA1P_M1_C1 + t * c;
c = INV_GAMMA1P_M1_C0 + t * c;
if (x > 0.5) {
ret = (t / x) * ((c - 0.5) - 0.5);
} else {
ret = x * c;
}
}
return ret;
}
/**
* Returns the value of log Γ(1 + x) for -0.5 ≤ x ≤ 1.5.
* This implementation is based on the double precision implementation in
* the NSWC Library of Mathematics Subroutines, {@code DGMLN1}.
*
* @param x Argument.
* @return The value of {@code log(Gamma(1 + x))}.
* @throws NumberIsTooSmallException if {@code x < -0.5}.
* @throws NumberIsTooLargeException if {@code x > 1.5}.
* @since 3.1
*/
public static double logGamma1p(final double x)
throws NumberIsTooSmallException, NumberIsTooLargeException {
if (x < -0.5) {
throw new NumberIsTooSmallException(x, -0.5, true);
}
if (x > 1.5) {
throw new NumberIsTooLargeException(x, 1.5, true);
}
return -FastMath.log1p(invGamma1pm1(x));
}
/**
* Returns the value of Γ(x). Based on the NSWC Library of
* Mathematics Subroutines double precision implementation,
* {@code DGAMMA}.
*
* @param x Argument.
* @return the value of {@code Gamma(x)}.
* @since 3.1
*/
public static double gamma(final double x) {
if ((x == FastMath.rint(x)) && (x <= 0.0)) {
return Double.NaN;
}
final double ret;
final double absX = FastMath.abs(x);
if (absX <= 20.0) {
if (x >= 1.0) {
/*
* From the recurrence relation
* Gamma(x) = (x - 1) * ... * (x - n) * Gamma(x - n),
* then
* Gamma(t) = 1 / [1 + invGamma1pm1(t - 1)],
* where t = x - n. This means that t must satisfy
* -0.5 <= t - 1 <= 1.5.
*/
double prod = 1.0;
double t = x;
while (t > 2.5) {
t -= 1.0;
prod *= t;
}
ret = prod / (1.0 + invGamma1pm1(t - 1.0));
} else {
/*
* From the recurrence relation
* Gamma(x) = Gamma(x + n + 1) / [x * (x + 1) * ... * (x + n)]
* then
* Gamma(x + n + 1) = 1 / [1 + invGamma1pm1(x + n)],
* which requires -0.5 <= x + n <= 1.5.
*/
double prod = x;
double t = x;
while (t < -0.5) {
t += 1.0;
prod *= t;
}
ret = 1.0 / (prod * (1.0 + invGamma1pm1(t)));
}
} else {
final double y = absX + LANCZOS_G + 0.5;
final double gammaAbs = SQRT_TWO_PI / x *
FastMath.pow(y, absX + 0.5) *
FastMath.exp(-y) * lanczos(absX);
if (x > 0.0) {
ret = gammaAbs;
} else {
/*
* From the reflection formula
* Gamma(x) * Gamma(1 - x) * sin(pi * x) = pi,
* and the recurrence relation
* Gamma(1 - x) = -x * Gamma(-x),
* it is found
* Gamma(x) = -pi / [x * sin(pi * x) * Gamma(-x)].
*/
ret = -FastMath.PI /
(x * FastMath.sin(FastMath.PI * x) * gammaAbs);
}
}
return ret;
}
}