Mathematical Tools for Physics
14—Complex Variables 433 if I extend the integration limits to the whole axis (times^1 / 2 ). 1 2 ∫ C 1 dz (a^2 +z^2 )^2 C 1 As ...
14—Complex Variables 434 depend on the relative sign ofaandb, because the change of variablesθ′=θ+πchanges the coefficient ofb w ...
14—Complex Variables 435 14.7 Branch Points Before looking at any more uses of the residue theorem, I have to return the the sub ...
14—Complex Variables 436 On the path labelled 2, angleθgoes from zero to 0 .6 + 4π, and √ r eiθ/^2 varies from 1 to 1. 25 e(2π+0 ...
14—Complex Variables 437 Now I sew the sheets together along these cuts. But, I sew the top edge from sheet #0 to the bottom edg ...
14—Complex Variables 438 are. Sideais the same as sidea. The same forb. When you have more complicated surfaces, arising from mo ...
14—Complex Variables 439 0 1 2 a b b c c a LogarithmHow about a logarithm? lnz= ln ( reiθ ) = lnr+iθ. There’s a branch point at ...
14—Complex Variables 440 going around the second branch point will introduce a second identical factor. As(−1)^2 = +1, then when ...
14—Complex Variables 441 2 C 1 C The fact that the logarithm goes to infinity at the origin doesn’t matter because it is such a ...
14—Complex Variables 442 Whichvalue ofln(−a)do I take? That answer is dictated by how I arrived at the point−awhen I pushed the ...
14—Complex Variables 443 Take the radius of the circle large enough that only the first term in the numerator and the first term ...
14—Complex Variables 444 By assumption,fis bounded,|f(z)|≤M. A basic property of complex numbers is that|u+v|≤|u|+|v|for any com ...
14—Complex Variables 445 Problems 14.1 Explicitly integratezndzaround the circle of radiusRcentered at the origin. The numbernis ...
14—Complex Variables 446 14.7 What is a Laurent series expansion aboutz= 0to at least four terms for sinz/z^4 ez/z^2 (1−z) What ...
14—Complex Variables 447 14.15 In the integration of Eq. ( 13 ) the contourC 2 had a bump into the upper C? half-plane, but the ...
14—Complex Variables 448 14.25 Evaluate the residues of these functions at their singularities. a,b, andcare distinct. Six answe ...
14—Complex Variables 449 14.32 Evaluate ∫∞ 0 dx lnx a^2 +x^2 (What happens if you consider(lnx)^2 ?) Ans:(πlna)/ 2 a 14.33 Evalu ...
14—Complex Variables 450 14.38 Evaluate ∫∞ 0 dx 1 a^3 +x^3 Ans: 2 π √ 3 / 9 a^2 14.39 Go back to problem3.45and find the branch ...
Fourier Analysis Fourier series allow you to expand a function on a finite interval as an infinite series of trigonometric funct ...
15—Fourier Analysis 452 Now I have to express this in terms of the explicit basis functions in order to manipulate it. When you ...
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