Complex Function Theory

Contents Preface 1. Complex numbers and their functions 1.1 Complex numbers 1.1.1 Definition of a complex number 1.1.2 Addition, subtraction, multiplication, and division of complex numbers 1.2 Complex plane 1.2.1 Complex plane and complex numbers 1.2.1.1 Line and circle on the two-dimensional plane 1.2.2 Polar representation of complex numbers 1.2.3 Euler's formula and the polar form of complex numbers 1.2.4 Exponentiation and nth root of a complex number 1.3 Sequence and series of complex numbers 1.3.1 Sequence and limit 1.3.2 Series and its convergence 2. Complex Functions and Holomorphy 2.1 Complex functions and their continuity 2.2 The differentiability and holomorphy of complex functions 2.2.1 Differentiation of complex functions 2.2.2 Differential formulae 2.2.3 Cauchy–Riemann relationship and derivatives of inverse functions 2.2.3.1 Cauchy–Riemann relation 2.2.3.2 Derivative of inverse function 2.2.4 Partial differentiation by z and partial differentiation by z 3. Elementary Functions 3.1 Point at infinity 3.2 Power series 3.2.1 Convergence of a power series 3.2.2 Convergence radius 3.3 Exponential functions, trigonometric functions, and hyperbolic functions 3.3.1 Exponential functions 3.3.2 Trigonometric functions and hyperbolic functions 3.4 Logarithmic Functions 3.4.1 Definition and principal values of logarithmic functions 3.4.2 Multi-valuedness of logarithmic function and Riemann surface 3.5 General exponential functions and multi-valuedness 3.5.1 Definition of general exponential function 3.5.2 Mapping of multi-valued function w = z1/n, and Riemann surface 3.6 Infinite product 3.6.1 Definition of infinite product, convergence, and divergence 3.6.2 Example of infinite product (infinite product representation of sin z, cos z) 4. Conformal Transformation 4.1 Definition of conformal transformation 4.2 Simple examples of conformal transformations 4.3 Linear transformations 4.3.1 Entire functions and rational functions 4.3.2 Linear fractional functions and linear transformations 4.3.3 Example of linear transformation (conformal transformation) 4.4 Harmonic functions and conformal transformations 4.4.1 Transformation of Laplace equation by conformal mapping 4.4.2 Harmonic functions in electromagnetism and fluid dynamics 4.4.2.1 Electromagnetism, electrostatic potential, and lines of electric force 4.4.2.2 Irrotational ow, velocity potential, and stream function 4.4.2.3 Solution of Laplace equation where boundaries exist 4.4.3 Application to electromagnetism 4.4.4 Application to fluid dynamics 4.4.4.1 The problem of flow with circulation 5. Singularities 5.1 Isolated singularity 5.1.1 Removable singularity 5.1.2 Poles 5.1.3 (Isolated) essential singularity 5.2 Accumulation singularity 5.3 Branch points 6. Complex Integrals 6.1 Closed Jordan curves and the shape of the holomorphic domain 6.2 Definition of Complex Integral 6.3 Basic property of complex integral 6.4 Cauchy's integral theorem 6.4.1 Cauchy's integral theorem 6.4.2 Indefinite integral and its holomorphy 6.4.3 Multi-valuedness of logarithmic functions and integration of 1/z 6.5 Residue 6.5.1 Definition of residue and residue theorem 6.5.2 Residue of the point at infinity 6.6 Application of complex integrals 6.6.1 Application of residue theorem (calculation of definite integrals) 6.6.2 Handling branch points of multi-valued functions in definite integrals 7. Cauchy’s Integral Formula and Power Series Expansion of Complex Functions 7.1 Cauchy's integral formula and theorems derived from it 7.1.1 Cauchy's integral formula 7.1.2 Maximum modulus principle and Liouville's theorem 7.1.3 Fundamental theorem of algebra 7.2 Cauchy's integral theorem and holomorphy 7.2.1 Goursat's theorem and Morera's theorem 7.2.2 Application of Goursat's theorem 7.3 Taylor expansion and Laurent expansion 7.3.1 Taylor expansion (power series expansion around a holomorphic point) 7.3.2 Laurent expansion 8. Advanced Complex Integrals 8.1 Topology and topological space 8.1.1 Topology 8.1.2 Equivalence relation of continuous mapping 8.1.3 Loops and fundamental groups 8.1.4 Homotopy and products of loops 8.1.4.1 Replacing base point 8.1.4.2 Simply-connected 8.1.5 Degree or rotation number 8.2 Index of a point concerning closed curve, and generalized residue theorem 8.2.1 Index of a point concerning closed curve (winding number) 8.2.2 Generalization of Cauchy's integral formula 8.2.3 Generalization of residue theorem 8.3 Application of residue theorem: Argument principle and Rouché's theorem 8.3.1 The orders of poles and zero points 8.3.2 Argument principle 8.3.3 Rouché's theorem 8.3.3.1 Rouché's theorem 8.3.3.2 Fundamental theorem of algebra 9. Analytic Continuation and Riemann Surface 9.1 Theorem of identity and reection principle 9.1.1 Theorem of identity 9.1.2 The reection principle 9.2 Analytic continuation and Riemann surfaces 9.2.1 Analytic continuation 9.2.1.1 Analytic continuation and its uniqueness 9.2.1.2 Weierstrass' analytic continuation 9.2.2 Monodromy theorem 9.2.3 Riemann surfaces 9.2.4 Natural boundary 10. Meromorphic Functions 10.1 Partial fraction expansion of a meromorphic function, and the infinite product representation of an entire function 10.1.1 Partial fraction expansion (decomposition) of a meromorphic function 10.1.2 Infinite product representation of entire function 10.2 Γ functions and B functions 10.2.1 Γ functions and their analytic continuation 10.2.2 B functions 10.2.3 Stirling's formula and asymptotic expansion of Γ function 10.2.3.1 Stirling's formula 10.2.3.2 Asymptotic expansion 11. Elliptic Integrals and Elliptic Functions 11.1 Elliptic Integrals 11.1.1 Definition of elliptic integral 11.1.2 Standard forms of elliptic integral 11.1.3 Example of elliptic integral 11.1.4 Properties of elliptic integral 11.1.5 Elliptic integral and Jacobi's elliptic functions 11.2 Elliptic Functions 11.2.1 Double period 11.2.1.1 Singly periodic functions 11.2.1.2 Double periodic function 11.2.1.3 Parallelogram produced by double period on Gauss plane: Fundamental period parallelogram 11.2.2 Definition of elliptic function 11.2.3 Basic properties of elliptic functions 11.2.4 Weierstrass δ function 11.2.4.1 Weierstrass ζ function 11.2.4.2 Definition of δ function 11.2.4.3 Relationship between ζ function and δ function 11.2.4.4 Properties of δ function 11.2.5 Jacobi's elliptic functions sn w, cn w, dn w 11.2.5.1 Definition of Jacobi's elliptic function 11.2.5.2 Properties of Jacobi's elliptic functions 12. Ordinary Differential Equations of Complex Variables 12.1 Differential equations and series solutions 12.1.1 nth-order linear ordinary differential equations for complex variables 12.1.2 Series solution around a regular point 12.1.3 Series solution around a regular singular point 12.1.3.1 Regular singular point 12.1.3.2 Series solution around a regular singular point: Case of n = 2 12.1.3.3 The case that, in Eq. (12.13), the difference between the indices ρ1 and ρ2 is neither 0 nor integer 12.1.4 Frobenius' method for solution around regular singular point 12.1.4.1 The case f0(ρ) = p0(ρ – ρ0)2, i.e. ρ1 = ρ2 = ρ0 12.1.4.2 The case f0(ρ) = p0(ρ – ρ1)(ρ – ρ1 + n), i.e. ρ1 – ρ2 = n > 0 (n: an integer) 12.1.5 Series solution around the point at infinity 12.1.5.1 Conditions for z = 1 to be a regular singular point 12.1.5.2 Conditions for z = 1 to be a regular point 12.2 Riemann's P function and Gauss hypergeometric functions 12.2.1 Fuchsian differential equations and Riemann's P function 12.2.1.1 Fuchsian differential equations and Fuchsian relation 12.2.1.2 When there are only three regular singular points in a finite domain: Papperitz's relation and Riemann's P Function 12.2.1.3 When a1, a2, ∞ are regular singular points 12.2.2 Gauss hypergeometric differential equations and hypergeometric functions 12.2.2.1 Gauss hypergeometric differential equations 12.2.2.2 Gauss hypergeometric function I. The solution around the regular singular point z = 0 12.2.2.3 Gauss hypergeometric functions II. Solution around regular singular point z = 1 12.2.2.4 Gauss hypergeometric functions III. Solution around regular singular point z = 1 12.2.2.5 Relations between hypergeometric functions and functions that we already know 12.2.3 Broad properties of hypergeometric differential equations 12.2.3.1 24 independent solutions (Kummer's transformation formula) 12.2.3.2 Analytic continuation and monodromy groups 12.2.3.3 Monodromy groups 12.2.4 Recurrence formulae of hypergeometric functions 12.3 Differential equations which have irregular singular points: Conuent hypergeometric functions 12.3.1 Kummer's conuent hypergeometric differential equations and conuent hypergeometric functions 12.3.2 Recurrence formulae of Kummer's conuent hypergeometric functions 12.3.3 Solutions around irregular singular points, and asymptotic expansions 13. Orthogonal Polynomials 13.1 Orthogonal polynomials in finite interval (a, b) 13.1.1 Definition of orthogonal polynomials 13.1.2 Definition of inner product and orthogonality of orthogonal polynomial Fn(x) 13.1.2.1 Definition of inner product 13.1.2.2 Orthogonality of Fn(x) and Fm(x) when m ≠ n 13.1.2.3 Polynomial Fn(x) of order n must be orthogonal with any arbitrary polynomial of order m (m ≤ n – 1) 13.2 Orthogonal polynomials in infinite interval (a, ∞) 13.2.1 Definition of orthogonal polynomials 13.2.2 Definition of inner product and orthogonality of polynomial Gn(x) 13.3 Orthogonal polynomials in infinite interval (–∞, ∞) 13.3.1 Definition of orthogonal polynomials: Hermite polynomial 13.3.2 Definition of inner product and orthogonality of Hermite polynomial Hn(x) 13.4 Differential equations satisfied by orthogonal polynomials 14. Functions Written With Hypergeometric Functions 14.1 Problems in physics: physical phenomena and partial differential equations in spherically-symmetric systems 14.1.1 Laplace operator by three-dimensional polar coordinates 14.1.2 Spherical harmonics 14.1.2.1 Differential equations which ϴ(θ) should satisfy 14.1.2.2 Solution when m = 0: Legendre polynomial 14.1.2.3 Solution when m =integer (m ≠ 0): Associated Legendre function 14.1.2.4 Spherical harmonics Yn,m(θ, ϕ) 14.2 Legendre differential equation and Legendre's function: Example of P function 14.2.1 Legendre differential equations 14.2.2 Legendre functions 14.2.2.1 Legendre functions of the first kind 14.2.2.2 Legendre functions of the second kind 14.2.3 Recurrence formulae of Legendre functions 14.3 Legendre polynomials 14.3.1 Legendre polynomials of the first kind 14.3.2 Legendre polynomials of the second kind 14.3.3 Orthogonal relations and normalization integrals of Legendre polynomials 14.3.4 Generating function of Legendre polynomial 14.4 Associated Legendre functions and associated Legendre polynomials 14.4.1 Associated Legendre's differential equations and associated Legendre functions 14.4.2 Recurrence formulae for associated Legendre functions 14.4.3 Associated Legendre polynomial 14.4.3.1 Definition of associated Legendre polynomial 14.4.3.2 Normalization orthogonal relation of associated Legendre polynomial 15. Functions Written With Confluent Hypergeometric Functions 15.1 Weber–Hermite differential equations and Hermite functions 15.1.1 Problems in physics: Harmonic oscillator 15.1.2 Weber–Hermite differential equations and series solutions 15.1.3 Hermite polynomials 15.1.4 Orthogonality and normalization integrals of Hermite polynomials 15.1.5 Generating function of Hermite polynomials 15.2 Laguerre differential equations and Laguerre functions 15.2.1 Problems in physics: Hydrogen atom 15.2.2 Laguerre differential equations or associated differential equations, and series solutions and polynomial solutions 15.2.3 Orthogonality and normalization integrals of associated Laguerre polynomials and other polynomials 15.2.4 Generating function of Laguerre polynomial 15.3 Bessel's differential equations and Bessel functions 15.3.1 Problems in physics: Oscillation of a circular drum 15.3.2 Bessel's differential equations and expressing solutions with conuent hypergeometric functions 15.3.3 Bessel functions of integer orders 15.3.4 Half-integer Bessel functions and spherical Bessel functions Bibliography Index Contents Preface 1. Complex numbers and their functions 1.1 Complex numbers 1.1.1 Definition of a complex number 1.1.2 Addition, subtraction, multiplication, and division of complex numbers 1.2 Complex plane 1.2.1 Complex plane and complex numbers 1.2.1.1 Line and circle on the two-dimensional plane 1.2.2 Polar representation of complex numbers 1.2.3 Euler's formula and the polar form of complex numbers 1.2.4 Exponentiation and nth root of a complex number 1.3 Sequence and series of complex numbers 1.3.1 Sequence and limit 1.3.2 Series and its convergence 2. Complex Functions and Holomorphy 2.1 Complex functions and their continuity 2.2 The differentiability and holomorphy of complex functions 2.2.1 Differentiation of complex functions 2.2.2 Differential formulae 2.2.3 Cauchy–Riemann relationship and derivatives of inverse functions 2.2.3.1 Cauchy–Riemann relation 2.2.3.2 Derivative of inverse function 2.2.4 Partial differentiation by z and partial differentiation by z 3. Elementary Functions 3.1 Point at infinity 3.2 Power series 3.2.1 Convergence of a power series 3.2.2 Convergence radius 3.3 Exponential functions, trigonometric functions, and hyperbolic functions 3.3.1 Exponential functions 3.3.2 Trigonometric functions and hyperbolic functions 3.4 Logarithmic Functions 3.4.1 Definition and principal values of logarithmic functions 3.4.2 Multi-valuedness of logarithmic function and Riemann surface 3.5 General exponential functions and multi-valuedness 3.5.1 Definition of general exponential function 3.5.2 Mapping of multi-valued function w = z1/n, and Riemann surface 3.6 Infinite product 3.6.1 Definition of infinite product, convergence, and divergence 3.6.2 Example of infinite product (infinite product representation of sin z, cos z) 4. Conformal Transformation 4.1 Definition of conformal transformation 4.2 Simple examples of conformal transformations 4.3 Linear transformations 4.3.1 Entire functions and rational functions 4.3.2 Linear fractional functions and linear transformations 4.3.3 Example of linear transformation (conformal transformation) 4.4 Harmonic functions and conformal transformations 4.4.1 Transformation of Laplace equation by conformal mapping 4.4.2 Harmonic functions in electromagnetism and fluid dynamics 4.4.2.1 Electromagnetism, electrostatic potential, and lines of electric force 4.4.2.2 Irrotational ow, velocity potential, and stream function 4.4.2.3 Solution of Laplace equation where boundaries exist 4.4.3 Application to electromagnetism 4.4.4 Application to fluid dynamics 4.4.4.1 The problem of flow with circulation 5. Singularities 5.1 Isolated singularity 5.1.1 Removable singularity 5.1.2 Poles 5.1.3 (Isolated) essential singularity 5.2 Accumulation singularity 5.3 Branch points 6. Complex Integrals 6.1 Closed Jordan curves and the shape of the holomorphic domain 6.2 Definition of Complex Integral 6.3 Basic property of complex integral 6.4 Cauchy's integral theorem 6.4.1 Cauchy's integral theorem 6.4.2 Indefinite integral and its holomorphy 6.4.3 Multi-valuedness of logarithmic functions and integration of 1/z 6.5 Residue 6.5.1 Definition of residue and residue theorem 6.5.2 Residue of the point at infinity 6.6 Application of complex integrals 6.6.1 Application of residue theorem (calculation of definite integrals) 6.6.2 Handling branch points of multi-valued functions in definite integrals 7. Cauchy’s Integral Formula and Power Series Expansion of Complex Functions 7.1 Cauchy's integral formula and theorems derived from it 7.1.1 Cauchy's integral formula 7.1.2 Maximum modulus principle and Liouville's theorem 7.1.3 Fundamental theorem of algebra 7.2 Cauchy's integral theorem and holomorphy 7.2.1 Goursat's theorem and Morera's theorem 7.2.2 Application of Goursat's theorem 7.3 Taylor expansion and Laurent expansion 7.3.1 Taylor expansion (power series expansion around a holomorphic point) 7.3.2 Laurent expansion 8. Advanced Complex Integrals 8.1 Topology and topological space 8.1.1 Topology 8.1.2 Equivalence relation of continuous mapping 8.1.3 Loops and fundamental groups 8.1.4 Homotopy and products of loops 8.1.4.1 Replacing base point 8.1.4.2 Simply-connected 8.1.5 Degree or rotation number 8.2 Index of a point concerning closed curve, and generalized residue theorem 8.2.1 Index of a point concerning closed curve (winding number) 8.2.2 Generalization of Cauchy's integral formula 8.2.3 Generalization of residue theorem 8.3 Application of residue theorem: Argument principle and Rouché's theorem 8.3.1 The orders of poles and zero points 8.3.2 Argument principle 8.3.3 Rouché's theorem 8.3.3.1 Rouché's theorem 8.3.3.2 Fundamental theorem of algebra 9. Analytic Continuation and Riemann Surface 9.1 Theorem of identity and reection principle 9.1.1 Theorem of identity 9.1.2 The reection principle 9.2 Analytic continuation and Riemann surfaces 9.2.1 Analytic continuation 9.2.1.1 Analytic continuation and its uniqueness 9.2.1.2 Weierstrass' analytic continuation 9.2.2 Monodromy theorem 9.2.3 Riemann surfaces 9.2.4 Natural boundary 10. Meromorphic Functions 10.1 Partial fraction expansion of a meromorphic function, and the infinite product representation of an entire function 10.1.1 Partial fraction expansion (decomposition) of a meromorphic function 10.1.2 Infinite product representation of entire function 10.2 Γ functions and B functions 10.2.1 Γ functions and their analytic continuation 10.2.2 B functions 10.2.3 Stirling's formula and asymptotic expansion of Γ function 10.2.3.1 Stirling's formula 10.2.3.2 Asymptotic expansion 11. Elliptic Integrals and Elliptic Functions 11.1 Elliptic Integrals 11.1.1 Definition of elliptic integral 11.1.2 Standard forms of elliptic integral 11.1.3 Example of elliptic integral 11.1.4 Properties of elliptic integral 11.1.5 Elliptic integral and Jacobi's elliptic functions 11.2 Elliptic Functions 11.2.1 Double period 11.2.1.1 Singly periodic functions 11.2.1.2 Double periodic function 11.2.1.3 Parallelogram produced by double period on Gauss plane: Fundamental period parallelogram 11.2.2 Definition of elliptic function 11.2.3 Basic properties of elliptic functions 11.2.4 Weierstrass δ function 11.2.4.1 Weierstrass ζ function 11.2.4.2 Definition of δ function 11.2.4.3 Relationship between ζ function and δ function 11.2.4.4 Properties of δ function 11.2.5 Jacobi's elliptic functions sn w, cn w, dn w 11.2.5.1 Definition of Jacobi's elliptic function 11.2.5.2 Properties of Jacobi's elliptic functions 12. Ordinary Differential Equations of Complex Variables 12.1 Differential equations and series solutions 12.1.1 nth-order linear ordinary differential equations for complex variables 12.1.2 Series solution around a regular point 12.1.3 Series solution around a regular singular point 12.1.3.1 Regular singular point 12.1.3.2 Series solution around a regular singular point: Case of n = 2 12.1.3.3 The case that, in Eq. (12.13), the difference between the indices ρ1 and ρ2 is neither 0 nor integer 12.1.4 Frobenius' method for solution around regular singular point 12.1.4.1 The case f0(ρ) = p0(ρ – ρ0)2, i.e. ρ1 = ρ2 = ρ0 12.1.4.2 The case f0(ρ) = p0(ρ – ρ1)(ρ – ρ1 + n), i.e. ρ1 – ρ2 = n > 0 (n: an integer) 12.1.5 Series solution around the point at infinity 12.1.5.1 Conditions for z = 1 to be a regular singular point 12.1.5.2 Conditions for z = 1 to be a regular point 12.2 Riemann's P function and Gauss hypergeometric functions 12.2.1 Fuchsian differential equations and Riemann's P function 12.2.1.1 Fuchsian differential equations and Fuchsian relation 12.2.1.2 When there are only three regular singular points in a finite domain: Papperitz's relation and Riemann's P Function 12.2.1.3 When a1, a2, ∞ are regular singular points 12.2.2 Gauss hypergeometric differential equations and hypergeometric functions 12.2.2.1 Gauss hypergeometric differential equations 12.2.2.2 Gauss hypergeometric function I. The solution around the regular singular point z = 0 12.2.2.3 Gauss hypergeometric functions II. Solution around regular singular point z = 1 12.2.2.4 Gauss hypergeometric functions III. Solution around regular singular point z = 1 12.2.2.5 Relations between hypergeometric functions and functions that we already know 12.2.3 Broad properties of hypergeometric differential equations 12.2.3.1 24 independent solutions (Kummer's transformation formula) 12.2.3.2 Analytic continuation and monodromy groups 12.2.3.3 Monodromy groups 12.2.4 Recurrence formulae of hypergeometric functions 12.3 Differential equations which have irregular singular points: Conuent hypergeometric functions 12.3.1 Kummer's conuent hypergeometric differential equations and conuent hypergeometric functions 12.3.2 Recurrence formulae of Kummer's conuent hypergeometric functions 12.3.3 Solutions around irregular singular points, and asymptotic expansions 13. Orthogonal Polynomials 13.1 Orthogonal polynomials in finite interval (a, b) 13.1.1 Definition of orthogonal polynomials 13.1.2 Definition of inner product and orthogonality of orthogonal polynomial Fn(x) 13.1.2.1 Definition of inner product 13.1.2.2 Orthogonality of Fn(x) and Fm(x) when m ≠ n 13.1.2.3 Polynomial Fn(x) of order n must be orthogonal with any arbitrary polynomial of order m (m ≤ n – 1) 13.2 Orthogonal polynomials in infinite interval (a, ∞) 13.2.1 Definition of orthogonal polynomials 13.2.2 Definition of inner product and orthogonality of polynomial Gn(x) 13.3 Orthogonal polynomials in infinite interval (–∞, ∞) 13.3.1 Definition of orthogonal polynomials: Hermite polynomial 13.3.2 Definition of inner product and orthogonality of Hermite polynomial Hn(x) 13.4 Differential equations satisfied by orthogonal polynomials 14. Functions Written With Hypergeometric Functions 14.1 Problems in physics: physical phenomena and partial differential equations in spherically-symmetric systems 14.1.1 Laplace operator by three-dimensional polar coordinates 14.1.2 Spherical harmonics 14.1.2.1 Differential equations which Θ(θ) should satisfy 14.1.2.2 Solution when m = 0: Legendre polynomial 14.1.2.3 Solution when m =integer (m ≠ 0): Associated Legendre function 14.1.2.4 Spherical harmonics Yn,m(θ, φ) 14.2 Legendre differential equation and Legendre's function: Example of P function 14.2.1 Legendre differential equations 14.2.2 Legendre functions 14.2.2.1 Legendre functions of the first kind 14.2.2.2 Legendre functions of the second kind 14.2.3 Recurrence formulae of Legendre functions 14.3 Legendre polynomials 14.3.1 Legendre polynomials of the first kind 14.3.2 Legendre polynomials of the second kind 14.3.3 Orthogonal relations and normalization integrals of Legendre polynomials 14.3.4 Generating function of Legendre polynomial 14.4 Associated Legendre functions and associated Legendre polynomials 14.4.1 Associated Legendre's differential equations and associated Legendre functions 14.4.2 Recurrence formulae for associated Legendre functions 14.4.3 Associated Legendre polynomial 14.4.3.1 Definition of associated Legendre polynomial 14.4.3.2 Normalization orthogonal relation of associated Legendre polynomial 15. Functions Written With Confluent Hypergeometric Functions 15.1 Weber–Hermite differential equations and Hermite functions 15.1.1 Problems in physics: Harmonic oscillator 15.1.2 Weber–Hermite differential equations and series solutions 15.1.3 Hermite polynomials 15.1.4 Orthogonality and normalization integrals of Hermite polynomials 15.1.5 Generating function of Hermite polynomials 15.2 Laguerre differential equations and Laguerre functions 15.2.1 Problems in physics: Hydrogen atom 15.2.2 Laguerre differential equations or associated differential equations, and series solutions and polynomial solutions 15.2.3 Orthogonality and normalization integrals of associated Laguerre polynomials and other polynomials 15.2.4 Generating function of Laguerre polynomial 15.3 Bessel's differential equations and Bessel functions 15.3.1 Problems in physics: Oscillation of a circular drum 15.3.2 Bessel's differential equations and expressing solutions with conuent hypergeometric functions 15.3.3 Bessel functions of integer orders 15.3.4 Half-integer Bessel functions and spherical Bessel functions Bibliography Index