Mathematical Methods for Physics and Engineering : A Comprehensive Guide

23.4 CLOSED-FORM SOLUTIONS so we can write y(x)=f(x)+ √ 2 πλ ̃y(x). (23.25) If we now take the Fourier transform of (23.25) but ...

INTEGRAL EQUATIONS Substituting (23.31) into (23.30) we find y(x)=f(x)+f ̃(x)+y(−x), but on changingxto−xand substituting back i ...

23.5 NEUMANN SERIES 23.5 Neumann series As mentioned above, most integral equations met in practice will not be of the simple fo ...

INTEGRAL EQUATIONS we may write thenth-order approximation as yn(x)=f(x)+ ∑n m=1 λm ∫b a Km(x, z)f(z)dz. (23.35) The solution to ...

23.6 Fredholm theory common ratioλ/3. Thus,provided|λ|<3, this infinite series converges to the value λ/(3−λ), and the soluti ...

INTEGRAL EQUATIONS Neumann series, which converge only if the condition (23.38) is satisfied. Thus the Fredholm method leads to ...

23.7 SCHMIDT–HILBERT THEORY Let us begin by considering the homogeneous integral equation y=λKy, where the integral operatorKhas ...

INTEGRAL EQUATIONS sides of (23.51) withyj, we find ∑ i ai〈yj|yi〉=λ ∑ i ai λi 〈yj|yi〉+λ〈yj|Kf〉. (23.52) Since the eigenfunctions ...

23.8 Exercises thus Hermitian. In order to solve this inhomogeneous equation using SH theory, however, we must first find the ei ...

INTEGRAL EQUATIONS 23.5 Solve forφ(x) the integral equation φ(x)=f(x)+λ ∫ 1 0 [( x y )n + (y x )n] φ(y)dy, wheref(x) is bounded ...

23.8 EXERCISES (b) Obtain the eigenvalues and eigenfunctions over the interval [0, 2 π]if K(x, y)= ∑∞ n=1 1 n cosnxcosny. 23.8 B ...

INTEGRAL EQUATIONS By examining the special casesx=0andx= 1, show that f(x)= 2 (e+3)(e+1) [(e+2)ex−ee−x]. 23.13 The operatorMis ...

23.9 Hints and answers 23.9 Hints and answers 23.1 Definey(−x)=y(x) and use the cosine Fourier transform inversion theorem; y(x) ...

24 Complex variables Throughout this book references have been made to results derived from the the- ory of complex variables. T ...

24.1 Functions of a complex variable students are more at ease with the former type of statement, despite its lack of precision, ...

COMPLEX VARIABLES We then find that f′(z) = lim ∆z→ 0 [ (z+∆z)^2 −z^2 ∆z ] = lim ∆z→ 0 [ (∆z)^2 +2z∆z ∆z ] = ( lim ∆z→ 0 ∆z ) +2 ...

24.2 The Cauchy–Riemann relations
Show that the functionf(z)=1/(1−z)is analytic everywhere except atz=1. Sincef(z) is given exp ...

COMPLEX VARIABLES Forfto be differentiable at the pointz, expressions (24.3) and (24.4) must be identical. It follows from equat ...

24.2 THE CAUCHY–RIEMANN RELATIONS Sincexandyare related tozand its complex conjugatez∗by x= 1 2 (z+z∗)andy= 1 2 i (z−z∗), (24.6) ...

COMPLEX VARIABLES whereiandjare the unit vectors along thex-andy-axes, respectively. A similar expression exists for∇v, the norm ...