Digamma function

Abstract: GENERALIZED GAMMA FUNCTION The Gamma Fun ction G(a) Definition of the Generalized Gamma Function Properties of the Generalized Gamma Function Mellin and Laplace Transforms Asymptotic Representations The Macdonald Function The Digamma Function y(x) Generalization of Psi (Digamma) Function Integral Representations of yb (a) Properties of the Generalized Psi Function Graphical and Tabular Representations THE GENERALIZED INCOMPLETE GAMMA FUNCTIONS The Incomplete Gamma Functions Definition of the Generalized Incomplete Gamma Functions Properties of the Incomplete Generalized Gamma Functions Convolution Representations and Laplace Transforms Connection with Other Special Functions KdF Functions and Incomplete Integrals Representation in terms of KdF Functions Reduction Formulas for F0:2 1 2:0 0[x,y] Integrals of Product of Bessel and Gamma Functions Asymptotic Representations Integral Representation for G(a,x b) Graphical and Tabular Representations THE FAMILY OF THE GAMMA FUNCTIONS The Family of Incomplete Gamma Functions The Generalized Error Functions The Generalized Exponential Integral function The Generalized Fresnel Integrals The Decomposition Functions The Extended Decomposition functions The E(u,v) and F(u,v) Functions The e(u) and f(u) Functions Graphical and Tabular Representations EXTENSION OF GENERALIZED INCOMPLETE GAMMA FUNCTIONS Introduction The Decomposition Formula Recurrence Relations Laplace and K-Transform Representation Parametric Differentiation and Integration Connection with Other Special Functions Integral Representations Differential Representations The Mellin Transform Representation EXTENDED BETA FUNCTION The Beta Function The Incomplete Beta Function The Beta Probability Distribution Definition of the Extended Beta Function Properties of the Extended Beta Function Integral Representations of the Extended Beta Function Conncection with Other Special Functions Representations in Terms of Whittaker functions Extended Incomplete Beta Function The Extended Beta Distribution Graphical and Tabular Representations EXTENDED INCOMPLETE GAMMA FUNCTIONS Introduction Definition of the Extended Incomplete Gamma Functions The Decomposition Formula Recurrence Formula Connection with Other Special Functions The H-Function Incomplete Fox H-Functions EXTENDED RIEMANN ZETA FUNCTIONS Introduction Bernoulli's Numbers and Polynomials The Zeta Function Zeros 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 Function Extended Hurwitz Zeta Functions Extended Hurwitz Formulae Further Remarks and Comments Graphical and Tabular Representations PHASE-CHANGE HEAT-TRANSFER Introduction Constant Temperature Boundary Conditions Convective Boundary Conditions Freezing of Tissues around a Capillary Tube Freezing of Binary Alloys Freezing Around an Impurity Numerical Methods for Phase-Change Problems HEAT CONDUCTION WITH TIME-DEPENDENT BOUNDARY CONDITIONS Introduction Time-Dependent Surface Temperatures Time-Dependent Surface Heat Fluxes Illustrative Example HEAT CONDUCTION DUE TO TIME-DEPENDENT LASER SOURCES Introduction Mathematical Formulation Some Cases of Practical Interest A UNIFIED APPROACH TO HEAT SOURCE PROBLEMS Introduction Thermal Explosions Continuously Operating Heat Sources APPENDICES Heat Conduction Table of Laplace Transforms Integrals Dependent of Parameters REFERENCES SYMBOLS INDEX

Abstract: The present paper starts with a systematic study of distribution functions in ^-dimensional space and in particular of their infinite convolutions representing, in the language of the calculus of probability, the distributions arising by addition of an infinite number of independent random variables. The results are applied to almost periodic functions and in particular to the Riemann zeta function. The proper method in dealing with distribution functions and their convolutions (\"Faltungen\") is the method of Fourier transforms, first applied systematically by Levy in his book on the calculus of probability [40].| The theorems concerning Fourier transforms which we need are collected at the beginning; for proofs we refer to papers of Bochner [2] and Haviland [28, 29]. These authors use Riemann-Radon integrals; we prefer for several reasons to work with Lebesgue-Radon integrals for which the proofs are simpler. Using these results on Fourier transforms we develop a general theory of infinite convolutions and in particular their convergence theory. This theory is completed at the end of the paper, where it is shown, by means of integrals in infinitely many dimensions, that the convergence problem of infinite convolutions is identical with the convergence problem of infinite series the terms of which are independent random variables, as considered by Khintchine and Kolmogoroff [37], Kolmogoroff [38], and L6vy [41]; incidentally we obtain a new treatment of the latter problem. The dominating feature of the convolution process is its smoothing effect, although it is hardly possible to formulate a single theorem covering the whole situation. In the cases in which we are interested an appraisal of the Fourier transforms is the natural approach to the treatment of the question. This method has recently been applied in the case of circular equidistributions by Wintner [55]; in the present paper it will be applied to the more general case of distributions on convex curves, fundamental for the treatment of the zeta function. The results thus obtained are essentially finer than those obtained through geometrical considerations by Bohr and Jessen [19]. These results are then applied to the almost periodic functions of Bohr

Abstract: : This article is concerned with the further development of the methods of p-adic analysis used in an earlier article to study the zeta function of an algebraic variety defined over a finite field. These methods are applied to the zeta function of a nonsingular hypersurface of degree d in projective n-space of characteristic p defined over the field of q elements.

Abstract: Preface.- The Constant Function c.- The Factorial Function n!.- The Zeta Numbers and Related Functions.- The Bernoulli Numbers Bn.- The Euler Numbers En.- The Bionmial Coefficients.- The Linear Function bx + c and Its Reciprocal.- Modifying Functions.- The Heaviside and Dirac Functions.- The Integer Powers xn and (bx + c)n.- The Square-Root Function and Its Reciprocal.- The Noninteger Power xv.- The Semielliptic Function and Its Reciprocal.- The (b/a)square root of x2 +- a2 Functions and Their Reciprocals.- The Quadratic Function ax + bx + c and Its Reciprocal.- The Cubic Function x3 + bx + c.- Polynomial Functions.- The Pochhammer Polynomials (x)n.- The Bernoulli Polynomials Bn(x).- The Euler Polynomials En(x).- The Legendre Polynomials Pn(x).- The Chebyshev Polynomials Tn(x) and Un(x).- The Laguerre Polynomials Ln(x).- The Hermite Polynomials Hn(x).- The Logarithmic Function ln(x).- The Exponential Function exp(x).- Exponential of Powers.- The Hyperbolic Cosine cosh(x). and Sine sinh(x) Functions.- The Hyperbolic Secant and Cosecant Functions.- The Inverse Hyperbolic Functions.- The Cosine cox(x) and Sine sin(x) Functions.- The Secant sec(x) and Cosecant csc(x) Fucntions.- The Tangent tan(x) and Cotangent cot(x) Functions.- The Inverse Circular Functions.- Periodic Functions.- The Exponential Integrals Ei(x) and Ein(x).- Sine and Cosine Integrals.- The Fresnel Integrals C(x) and S(x).- The Error Function erf(x) and Its Complement erfc(x).- The exp(x)erfc(square root of x) and Related Functions.- Dawson's Integral daw(x).- The Gamma Function.- The Digamma Function.- The Incomplete Gamma Functions.- The Parabolic Cylinder Function Dv(x).- The Kummer Function M(a, c, x).- The Tricomi Function U(a, c, x).- The Modified Bessel Functions In(x) of Integer Order.- The Modified Bessel Functions of In(x) Arbitrary Order.- The Macdonald Function Kv(x).- The Bessel Functions Jn(x) of Integer Order.- The Bessel Functions Jv(x) of Arbitrary Order.- The Neumann Function Yv(x). The Kelvin Functions.- The Airy Functions Ai(x) and Bi(x).- The Struve Function hv(x).- The Incomplete Beta Function.- The Legendre Functions Pv(x) and Qv(x).- The Gauss Hypergeometric Function F(a,b,c,x).- The Complete Elliptic Integrals K(k) and E(k).- The Incomplete Elliptic Integrals.- The Jacobian Elliptic Functions.- The Hurwitz Function.- Appendix A: Useful Data.- Appendix B: Bibliography.- Appendix C: Equator, The Atlas Function Calculator.- Symbol Index.- Subject Index.