Mathematical Tools for Physics - Department of Physics - University

14—Complex Variables 354 The lower limit may be finite, but that just makes it easier. In problem14.1you found that the integral ...

14—Complex Variables 355 as long as it doesn’t move across a singularity, you can distort a contour at will. I will push the con ...

14—Complex Variables 356 zero. It goes to zero faster than any inverse power ofy, so even with the length of the contour going a ...

14—Complex Variables 357 because it applies only for the case thatk > 0. If you have a negativekyou can do the integral again ...

14—Complex Variables 358 1 2 1 (a^2 +z^2 )^2 = 1 2 1 (z−ia)^2 (z+ia)^2 = 1 2 1 (z−ia)^2 (z−ia+ 2ia)^2 = 1 2 1 (z−ia)^2 (2ia)^2 [ ...

14—Complex Variables 359 Asλvaries from 1 to∞, the two roots travel from− 1 → −∞and from− 1 → 0 , soz+stays inside the unit circ ...

14—Complex Variables 360 1 There are two classes of paths fromz 0 toz, those that go around the z origin an even number of times ...

14—Complex Variables 361 b. When you have more complicated surfaces, arising from more complicated functions of the complex vari ...

14—Complex Variables 362 LogarithmHow about a logarithm?lnz= ln ( reiθ ) = lnr+iθ. There’s a branch point at the origin, but thi ...

14—Complex Variables 363 Example 8 The integral ∫∞ 0 dxx/(a+x) (^3). You can do this by elementary methods (very easily in fact) ...

14—Complex Variables 364 Whichvalue ofln(−a)to take? That answer is dictated by how I arrived at the point−awhen I pushed the co ...

14—Complex Variables 365 Eq. (14.18) is Cauchy’s integral formula, giving the analytic function in terms of its boundary values. ...

14—Complex Variables 366 Problems 14.1Explicitly integratezndzaround the circle of radiusRcentered at the origin, just as in Eq. ...

14—Complex Variables 367 14.9 Evaluate the integral along the straight line fromatoa+i∞: ∫ eizdz. Takeato be real. Ans:ieia 14.1 ...

14—Complex Variables 368 whereCis a circle of radiusπnabout the origin. Ans:− 4 πin 14.25 Evaluate the residues of these functio ...

14—Complex Variables 369 14.35 Evaluate the integral of problem14.33another way. Assumeλis large and expand the integrand in a p ...

Fourier Analysis . Fourier series allow you to expand a function on a finite interval as an infinite series of trigonometric fun ...

15—Fourier Analysis 371 For a given value ofk, define the integral gL(k) = ∫L −L dx′e−ikx ′ f(x′) If the functionfvanishes suffi ...

15—Fourier Analysis 372 1. Ifx >+athen bothx+aandx−aare positive, which implies that both exponentials vanish rapidly ask→+i∞ ...

15—Fourier Analysis 373 = ∫ dk′ 2 π g 1 (k′) ∫ dxeik ′x f 2 (x)e−ikx = ∫ dk′ 2 π g 1 (k′) ∫ dxf 2 (x)e−i(k−k ′)x = ∫∞ −∞ dk′ 2 π ...