On a Class of Incomplete Gamma Functions with Applications

ISBN-13: 9781584881438 / Angielski / Twarda / 2001 / 512 str.

M. Aslam Chaudhry; Syed M. Zubair

Opis

The subject of special functions is rich and expanding continuously with the emergence of new problems encountered in engineering and applied science applications. The development of computational techniques and the rapid growth in computing power have increased the importance of the special functions and their formulae for analytic representations. However, problems remain, particularly in heat conduction, astrophysics, and probability theory, whose solutions seem to defy even the most general classes of special functions.
On a Class of Incomplete Gamma Functions with Applications introduces a class of special functions, developed by the authors, useful in the analytic study of several heat conduction problems. It presents some basic properties of these functions, including their recurrence relations, special cases, asymptotic representations, and integral transform relationships. The authors explore applications of these generalized functions to problems in transient heat conduction, special cases of laser sources, and problems associated with heat transfer in human tissues. They also discuss applications to astrophysics, probability theory, and other problems in theory of functions and present a fundamental solution to time-dependent laser sources with convective-type boundary conditions.
Appendices include an introduction to heat conduction, Fourier conduction, a table of Laplace transforms, and well-known results regarding the improper integrals. Filled with tabular and graphical representations for applications, this monograph offers a unique opportunity to add to your mathematical toolbox a new and useful class of special functions.

Spis treści

GENERALIZED GAMMA FUNCTIONThe Gamma Fun ction G(a)Definition of the Generalized Gamma FunctionProperties of the Generalized Gamma FunctionMellin and Laplace TransformsAsymptotic RepresentationsThe Macdonald FunctionThe Digamma Function y(x)Generalization of Psi (Digamma) FunctionIntegral Representations of yb (a)Properties of the Generalized Psi FunctionGraphical and Tabular RepresentationsTHE GENERALIZED INCOMPLETE GAMMA FUNCTIONSThe Incomplete Gamma FunctionsDefinition of the Generalized Incomplete Gamma FunctionsProperties of the Incomplete Generalized Gamma FunctionsConvolution Representations and Laplace TransformsConnection with Other Special FunctionsKdF Functions and Incomplete IntegralsRepresentation in terms of KdF FunctionsReduction Formulas for F0:2;1 2:0;0[x,y]Integrals of Product of Bessel and Gamma FunctionsAsymptotic RepresentationsIntegral Representation for G(a,x;b)Graphical and Tabular RepresentationsTHE FAMILY OF THE GAMMA FUNCTIONSThe Family of Incomplete Gamma FunctionsThe Generalized Error FunctionsThe Generalized Exponential Integral functionThe Generalized Fresnel IntegralsThe Decomposition FunctionsThe Extended Decomposition functionsThe E(u,v) and F(u,v) FunctionsThe e(u) and f(u) FunctionsGraphical and Tabular RepresentationsEXTENSION OF GENERALIZED INCOMPLETE GAMMA FUNCTIONSIntroductionThe Decomposition FormulaRecurrence RelationsLaplace and K-Transform RepresentationParametric Differentiation and IntegrationConnection with Other Special FunctionsIntegral RepresentationsDifferential RepresentationsThe Mellin Transform RepresentationEXTENDED BETA FUNCTIONThe Beta FunctionThe Incomplete Beta FunctionThe Beta Probability DistributionDefinition of the Extended Beta FunctionProperties of the Extended Beta FunctionIntegral Representations of the Extended Beta FunctionConncection with Other Special FunctionsRepresentations in Terms of Whittaker functionsExtended Incomplete Beta FunctionThe Extended Beta DistributionGraphical and Tabular RepresentationsEXTENDED INCOMPLETE GAMMA FUNCTIONSIntroductionDefinition of the Extended Incomplete Gamma FunctionsThe Decomposition FormulaRecurrence FormulaConnection with Other Special FunctionsThe H-FunctionIncomplete Fox H-FunctionsEXTENDED RIEMANN ZETA FUNCTIONSIntroductionBernoulli's Numbers and PolynomialsThe Zeta FunctionZeros of the Zeta Function and the Function p(x)The Extended Zeta Function z(a)The Second Extended Zeta Function zb*(a)The Hurwitz Zeta FunctionExtended Hurwitz Zeta FunctionsExtended Hurwitz FormulaeFurther Remarks and CommentsGraphical and Tabular RepresentationsPHASE-CHANGE HEAT-TRANSFERIntroductionConstant Temperature Boundary ConditionsConvective Boundary ConditionsFreezing of Tissues around a Capillary TubeFreezing of Binary AlloysFreezing Around an ImpurityNumerical Methods for Phase-Change ProblemsHEAT CONDUCTION WITH TIME-DEPENDENT BOUNDARY CONDITIONSIntroductionTime-Dependent Surface TemperaturesTime-Dependent Surface Heat FluxesIllustrative ExampleHEAT CONDUCTION DUE TO TIME-DEPENDENT LASER SOURCESIntroductionMathematical FormulationSome Cases of Practical InterestA UNIFIED APPROACH TO HEAT SOURCE PROBLEMSIntroductionThermal ExplosionsContinuously Operating Heat SourcesAPPENDICESHeat ConductionTable of Laplace TransformsIntegrals Dependent of ParametersREFERENCESSYMBOLSINDEX

O autorze

Syed M. Zubair, M. Aslam Chaudhry