Mathematical Physics Vol 1

5.4.1 Besselequation...............................23.4 5.4.2 Weberfunctions ..............................2.41 5.5 Someotherspecialfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . .24.3 5.5.1 Hermitpolynomials ............................24.3 5.5.2 Laguerrepolynomials ...........................24.4 5.6 Special functions that are not a result of the Frobenius method . . . . . . . . . .24. 4 5.6.1 Gamma function (factorial function) . . . . . . . . . . . . . . . . . . . .24. 4 5.6.2 Betafunction ................................2.51 5.6.3 Errorfunction................................25.3 5.6.4 Exponentialintegrals............................25.4 5.7 Mittag-Lefflerfunctions...............................25.5 5.8 Ellipticintegrals...................................25.7 5.8.1 Some properties of the integral F ( ϕ , k ) ..................25.8 5.8.2 Ellipticfunctions ..............................25.9 5.8.3 Complete elliptic integrals of the first and second kind . . . . . . . . . .26. 0 5.8.4 Jacobiellipticfunctions . . . . . . . . . . . . . . . . . . . . . . . . . .26.0 5.8.5 Main properties of elliptic functions . . . . . . . . . . . . . . . . . . . .2.61 5.9 Orthogonal and normalized functions . . . . . . . . . . . . . . . . . . . . . . .26. 3 5.9.1 Series of orthogonal functions . . . . . . . . . . . . . . . . . . . . . . .26. 5 5.9.2 Completeness of orthonormal functions . . . . . . . . . . . . . . . . . .26. 5 5.9.3 Sturm–Liouvilleproblem. . . . . . . . . . . . . . . . . . . . . . . . . .26.6 5.10Examples ......................................27.0 307 6.1 Periodicfunctions..................................30.7 6.1.1 Properties of periodic functions . . . . . . . . . . . . . . . . . . . . . .30. 8 6.1.2 Extension of non-periodic functions . . . . . . . . . . . . . . . . . . . .30. 9 6.1.3 Sum (superposition) of harmonics . . . . . . . . . . . . . . . . . . . . .30. 9 6.2 The fundamental convergence theorem for Fourier series . . . . . . . . . . . . .3.11 6.2.1 Expanding even and odd functions into Fourier series. Fourier sine and cosineseries ................................31.2 6.2.2 Expansion of functions into Fourier series on the interval ( − π , π ) . . . .31. 5 6.2.3 Expansion of functions into Fourier series on the interval ( 0 ,ℓ ) . Extensionofthehalf-interval. . . . . . . . . . . . . . . . . . . . . . . .31.5 6.2.4 Approximation of a function by a trigonometric polynomial. Mean squareerror.................................31.7 6.2.5 Complex form of Fourier series . . . . . . . . . . . . . . . . . . . . . .3.21 6.2.6 Fourierintegral...............................32.2 6.3 Examples ......................................32.4 Trigonometric Fourier series. Fourier integral

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305 Chapter 6 Trigonometric Fourier series. Fourier integral

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PDE

331 Chapter 7 Partial differential equations

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