BesselSecondEqualityLemma.java
package org.drip.specialfunction.property;
/*
* -*- mode: java; tab-width: 4; indent-tabs-mode: nil; c-basic-offset: 4 -*-
*/
/*!
* Copyright (C) 2020 Lakshmi Krishnamurthy
* Copyright (C) 2019 Lakshmi Krishnamurthy
*
* This file is part of DROP, an open-source library targeting analytics/risk, transaction cost analytics,
* asset liability management analytics, capital, exposure, and margin analytics, valuation adjustment
* analytics, and portfolio construction analytics within and across fixed income, credit, commodity,
* equity, FX, and structured products. It also includes auxiliary libraries for algorithm support,
* numerical analysis, numerical optimization, spline builder, model validation, statistical learning,
* and computational support.
*
* https://lakshmidrip.github.io/DROP/
*
* DROP is composed of three modules:
*
* - DROP Product Core - https://lakshmidrip.github.io/DROP-Product-Core/
* - DROP Portfolio Core - https://lakshmidrip.github.io/DROP-Portfolio-Core/
* - DROP Computational Core - https://lakshmidrip.github.io/DROP-Computational-Core/
*
* DROP Product Core implements libraries for the following:
* - Fixed Income Analytics
* - Loan Analytics
* - Transaction Cost Analytics
*
* DROP Portfolio Core implements libraries for the following:
* - Asset Allocation Analytics
* - Asset Liability Management Analytics
* - Capital Estimation Analytics
* - Exposure Analytics
* - Margin Analytics
* - XVA Analytics
*
* DROP Computational Core implements libraries for the following:
* - Algorithm Support
* - Computation Support
* - Function Analysis
* - Model Validation
* - Numerical Analysis
* - Numerical Optimizer
* - Spline Builder
* - Statistical Learning
*
* Documentation for DROP is Spread Over:
*
* - Main => https://lakshmidrip.github.io/DROP/
* - Wiki => https://github.com/lakshmiDRIP/DROP/wiki
* - GitHub => https://github.com/lakshmiDRIP/DROP
* - Repo Layout Taxonomy => https://github.com/lakshmiDRIP/DROP/blob/master/Taxonomy.md
* - Javadoc => https://lakshmidrip.github.io/DROP/Javadoc/index.html
* - Technical Specifications => https://github.com/lakshmiDRIP/DROP/tree/master/Docs/Internal
* - Release Versions => https://lakshmidrip.github.io/DROP/version.html
* - Community Credits => https://lakshmidrip.github.io/DROP/credits.html
* - Issues Catalog => https://github.com/lakshmiDRIP/DROP/issues
* - JUnit => https://lakshmidrip.github.io/DROP/junit/index.html
* - Jacoco => https://lakshmidrip.github.io/DROP/jacoco/index.html
*
* Licensed 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.
*/
/**
* <i>BesselSecondEqualityLemma</i> implements the implements the Equality Lemmas for the Cylindrical Bessel
* Function of the Second Kind. The References are:
*
* <br><br>
* <ul>
* <li>
* Abramowitz, M., and I. A. Stegun (2007): <i>Handbook of Mathematics Functions</i> <b>Dover Book
* on Mathematics</b>
* </li>
* <li>
* Arfken, G. B., and H. J. Weber (2005): <i>Mathematical Methods for Physicists 6<sup>th</sup>
* Edition</i> <b>Harcourt</b> San Diego
* </li>
* <li>
* Temme N. M. (1996): <i>Special Functions: An Introduction to the Classical Functions of
* Mathematical Physics 2<sup>nd</sup> Edition</i> <b>Wiley</b> New York
* </li>
* <li>
* Watson, G. N. (1995): <i>A Treatise on the Theory of Bessel Functions</i> <b>Cambridge University
* Press</b>
* </li>
* <li>
* Wikipedia (2019): Bessel Function https://en.wikipedia.org/wiki/Bessel_function
* </li>
* </ul>
*
* <br><br>
* <ul>
* <li><b>Module </b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/ComputationalCore.md">Computational Core Module</a></li>
* <li><b>Library</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/FunctionAnalysisLibrary.md">Function Analysis Library</a></li>
* <li><b>Project</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/specialfunction/README.md">Special Function Implementation Analysis</a></li>
* <li><b>Package</b> = <a href = "https://github.com/lakshmiDRIP/DROP/tree/master/src/main/java/org/drip/specialfunction/property/README.md">Special Function Property Lemma Verifiers</a></li>
* </ul>
*
* @author Lakshmi Krishnamurthy
*/
public class BesselSecondEqualityLemma
{
/**
* Construct the Bessel Second Kind Mirror Identity Verifier
*
* @return The Bessel Second Kind Mirror Identity Verifier
*/
public static final org.drip.function.definition.R2ToR1Property MirrorIdentity()
{
try
{
org.drip.function.definition.R1ToR1 gammaEstimator = new
org.drip.specialfunction.gamma.EulerIntegralSecondKind (null);
final org.drip.specialfunction.definition.BesselSecondKindEstimator besselSecondKindEstimator =
org.drip.specialfunction.bessel.SecondNISTSeriesEstimator.Standard (
new org.drip.specialfunction.digamma.BinetFirstIntegral (null),
gammaEstimator,
org.drip.specialfunction.bessel.FirstFrobeniusSeriesEstimator.Standard (
gammaEstimator,
40
),
40
);
return new org.drip.function.definition.R2ToR1Property (
org.drip.function.definition.RxToR1Property.EQ,
new org.drip.function.definition.R2ToR1()
{
@Override public double evaluate (
final double alpha,
final double z)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsInteger (alpha))
{
throw new java.lang.Exception
("BesselFirstEqualityLemma::MirrorIdentity => Invalid Inputs");
}
return besselSecondKindEstimator.bigY (
-1. * alpha,
z
);
}
},
new org.drip.function.definition.R2ToR1()
{
@Override public double evaluate (
final double alpha,
final double z)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsInteger (alpha))
{
throw new java.lang.Exception
("BesselFirstEqualityLemma::MirrorIdentity => Invalid Inputs");
}
return (0 == ((int) z) % 2 ? 1. : -1.) * besselSecondKindEstimator.bigY (
alpha,
z
);
}
},
org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
/**
* Construct the Bessel Second Kind Half-Integer Identity Verifier
*
* @return The Bessel Second Kind Half-Integer Identity Verifier
*/
public static final org.drip.function.definition.R2ToR1Property HalfIntegerIdentity()
{
org.drip.function.definition.R1ToR1 gammaEstimator = new
org.drip.specialfunction.gamma.EulerIntegralSecondKind (null);
final org.drip.specialfunction.definition.BesselFirstKindEstimator besselFirstKindEstimator =
org.drip.specialfunction.bessel.FirstFrobeniusSeriesEstimator.Standard (
gammaEstimator,
50
);
try
{
final org.drip.specialfunction.definition.BesselSecondKindEstimator besselSecondKindEstimator =
org.drip.specialfunction.bessel.SecondNISTSeriesEstimator.Standard (
new org.drip.specialfunction.digamma.BinetFirstIntegral (null),
gammaEstimator,
org.drip.specialfunction.bessel.FirstFrobeniusSeriesEstimator.Standard (
gammaEstimator,
40
),
40
);
return new org.drip.function.definition.R2ToR1Property (
org.drip.function.definition.RxToR1Property.EQ,
new org.drip.function.definition.R2ToR1()
{
@Override public double evaluate (
final double alpha,
final double z)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsInteger (alpha))
{
throw new java.lang.Exception
("BesselSecondEqualityLemma::HalfIntegerIdentity => Invalid Inputs");
}
return besselSecondKindEstimator.bigY (
-1. * (alpha + 0.5),
z
);
}
},
new org.drip.function.definition.R2ToR1()
{
@Override public double evaluate (
final double alpha,
final double z)
throws java.lang.Exception
{
if (!org.drip.numerical.common.NumberUtil.IsInteger (alpha))
{
throw new java.lang.Exception
("BesselSecondEqualityLemma::HalfIntegerIdentity => Invalid Inputs");
}
return (0 == ((int) (alpha)) % 2 ? 1. : -1.) * besselFirstKindEstimator.bigJ (
alpha + 0.5,
z
);
}
},
org.drip.function.definition.R1ToR1Property.MISMATCH_TOLERANCE
);
}
catch (java.lang.Exception e)
{
e.printStackTrace();
}
return null;
}
}