# Under such conditions show that i α πα 1 α 2 sin

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Under such conditions, show that I ( α ) = πα (1 α ) 2 sin πα . Show that I is continuous at 0 and 1. Q 10.16. (Q7(a), Paper II, 1994) The function h ( t ) vanishes for t< 0. The integral P ( t ) = integraldisplay t −∞ h ( t τ ) f ( τ ) has the property that df dt = integraldisplay t −∞ h ( t τ ) P ( τ ) for all well behaved f . Find ˆ h ( ω ) 2 . Q 10.17. (Q7, Paper II, 1998) Suppose that ˆ f ( ω ) = e e Find f by the following two methods. (i) By using formulae for such things as the Fourier transforms of deriva- tives, translates and Heaviside type functions together with the uniqueness of Fourier transforms. (ii) Directly from the inversion formula. [ Hint: You will need to distinguish between, t< 1 , 1 <t< 1 and t> 1 . ] Q 10.18. (Q16, Paper II, 1998) Use Fourier transform methods to solve the following integral equation for f ( t ), f ( t ) + integraldisplay 0 e s f ( t s ) ds = braceleftBigg e t if t 0, 0 if t< 0. Evaluate the convolution integral for your solution and hence confirm that f ( t ) solves the integral equation in the form stated above. 28

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Q 10.19. (Q8(a), Paper IV, 1994) The analytic function f ( z ) has P poles and Z zeros. All the poles and zeros lie strictly within the smooth, non-self- intersecting curve C . Using Cauchy’s integral formulas, show that, if all the poles and zeros are simple, integraldisplay C f ( z ) f ( z ) dz = 2 π ( Z P ) . Explain how your result must be modified if the poles and zeros are not simple and prove the modified result. Restate your result in terms of the argument of f . Part B Q 10.20. Cauchy gave the following example of a well behaved real function with no useful Taylor expansion about 0. It is important that you work through it at some stage in your mathematical life. Let E : R R be defined by E ( t ) = exp( 1 /t 2 ) for t negationslash = 0 and E (0). Use induction to show that E is infinitely differentiable with E ( n ) ( t ) = Q n (1 /t ) E ( t ) for t negationslash = 0 , E ( n ) (0) = 0 . For which values of t is it true that E ( t ) = summationdisplay n =0 E n (0) t n n ! ? Why does this not contradict Theorem 5.4? Q 10.21. (Q7, Paper I, 1999) For each of the following five functions, state the region of the complex plane in which it is complex differentiable. State also the region in which it has partial derivatives and the region in which they satisfy the Cauchy-Riemann conditions. f 1 ( z ) = | z | , f 2 ( z ) = e z , f 3 ( z ) = z , f 4 ( z ) = ( z 1) 3 , f 5 ( z ) = | z | 2 29
and f 6 given by f 6 ( x + iy ) = braceleftBigg xy ( x 2 + y 2 ) 1 / 2 if x + iy negationslash = 0, 0 if x + iy = 0. [ Note that, in one case, the region of complex differentiability does not coin- cide with that of the validity of the Cauchy-Riemann equations. ] Q 10.22. (Q17, Paper IV, 1999) Write down the Cauchy-Riemann equations for the real and imaginary parts of the analytic function w ( z ) = u ( x,y ) + iv ( x,y ), where z = x + iy . Show that 2 u = 2 v = 0 (i.e. that u and v are harmonic functions). Prove also that the curves of constant u in the x,y plane intersect those of constant v orthogonally.

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