LEADER 05044nam 2200637 a 450 001 9910841474403321 005 20230421050529.0 010 $a1-280-76794-4 010 $a9786613678713 010 $a1-118-03257-8 010 $a1-118-03081-8 035 $a(CKB)2550000000031175 035 $a(EBL)675156 035 $a(OCoLC)710974986 035 $a(SSID)ssj0000487529 035 $a(PQKBManifestationID)11309495 035 $a(PQKBTitleCode)TC0000487529 035 $a(PQKBWorkID)10442626 035 $a(PQKB)11072362 035 $a(MiAaPQ)EBC675156 035 $a(EXLCZ)992550000000031175 100 $a19950920d1996 uy 0 101 0 $aeng 135 $aur|n|---||||| 181 $ctxt 182 $cc 183 $acr 200 10$aSpecial functions$b[electronic resource] $ean introduction to the classical functions of mathematical physics /$fNico M. Temme 210 $aNew York $cWiley$d1996 215 $a1 online resource (392 p.) 300 $a"A Wiley-Interscience publication"--t.p. 311 $a0-471-11313-1 320 $aIncludes bibliographical references (p. 349-360) and index. 327 $aSpecial Functions: An Introduction to the Classical Functions of Mathematical Physics; Contents; 1 Bernoulli, Euler and Stirling Numbers; 1.1. Bernoulli Numbers and Polynomials; 1.1.1. Definitions and Properties; 1.1.2. A Simple Difference Equation; 1.1.3. Euler's Summation Formula; 1.2. Euler Numbers and Polynomials; 1.2.1. Definitions and Properties; 1.2.2. Boole's Summation Formula; 1.3. Stirling Numbers; 1.4. Remarks and Comments for Further Reading; 1.5. Exercises and Further Examples; 2 Useful Methods and Techniques; 2.1. Some Theorems from Analysis 327 $a2.2. Asymptotic Expansions of Integrals2.2.1. Watson's Lemma; 2.2.2. The Saddle Point Method; 2.2.3. Other Asymptotic Methods; 2.3. Exercises and Further Examples; 3 The Gamma Function; 3.1. Introduction; 3.1.1. The Fundamental Recursion Property; 3.1.2. Another Look at the Gamma Function; 3.2. Important Properties; 3.2.1. Prym's Decomposition; 3.2.2. The Cauchy-Saalschu?tz Representation; 3.2.3. The Beta Integral; 3.2.4. The Multiplication Formula; 3.2.5. The Reflection Formula; 3.2.6. The Reciprocal Gamma Function; 3.2.7. A Complex Contour for the Beta Integral; 3.3. Infinite Products 327 $a3.3.1. Gauss' Multiplication Formula3.4. Logarithmic Derivative of the Gamma Function; 3.5. Riemann's Zeta Function; 3.6. Asymptotic Expansions; 3.6.1. Estimations of the Remainder; 3.6.2. Ratio of Two Gamma Functions; 3.6.3. Application of the Saddle Point Method; 3.7. Remarks and Comments for Further Reading; 3.8. Exercises and Further Examples; 4 Differential Equations; 4.1. Separating the Wave Equation; 4.1.1. Separating the Variables; 4.2. Differential Equations in the Complex Plane; 4.2.1. Singular Points; 4.2.2. Transformation of the Point at Infinity 327 $a4.2.3. The Solution Near a Regular Point4.2.4. Power Series Expansions Around a Regular Point; 4.2.5. Power Series Expansions Around a Regular Singular Point; 4.3. Sturm's Comparison Theorem; 4.4. Integrals as Solutions of Differential Equations; 4.5. The Liouville Transformation; 4.6. Remarks and Comments for Further Reading; 4.7. Exercises and Further Examples; 5 Hypergeometric Functions; 5.1. Definitions and Simple Relations; 5.2. Analytic Continuation; 5.2.1. Three Functional Relations; 5.2.2. A Contour Integral Representation; 5.3. The Hypergeometric Differential Equation 327 $a5.4. The Singular Points of the Differential Equation5.5. The Riemann-Papperitz Equation; 5.6. Barnes' Contour Integral for F(a, b; c; z); 5.7. Recurrence Relations; 5.8. Quadratic Transformations; 5.9. Generalized Hypergeometric Functions; 5.9.1. A First Introduction to q-functions; 5.10. Remarks and Comments for Further Reading; 5.11. Exercises and Further Examples; 6 Orthogonal Polynomials; 6.1. General Orthogonal Polynomials; 6.1.1. Zeros of Orthogonal Polynomials; 6.1.2. Gauss Quadrature; 6.2. Classical Orthogonal Polynomials; 6.3. Definitions by the Rodrigues Formula 327 $a6.4. Recurrence Relations 330 $aThis book gives an introduction to the classical, well-known special functions which play a role in mathematical physics, especially in boundary value problems. Calculus and complex function theory form the basis of the book and numerous formulas are given. Particular attention is given to asymptomatic and numerical aspects of special functions, with numerous references to recent literature provided. 606 $aFunctions, Special 606 $aBoundary value problems 606 $aMathematical physics 615 0$aFunctions, Special. 615 0$aBoundary value problems. 615 0$aMathematical physics. 676 $a515.5 676 $a530.15 700 $aTemme$b N. M$0513107 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910841474403321 996 $aSpecial functions$92574162 997 $aUNINA