Conformal maps. Lent 2019 COMPLEX METHODS G. Taylor. A star means optional and not necessarily harder.

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1 Lent 29 COMPLEX METHODS G. Taylor A star means optional and not necessarily harder. Conformal maps. (i) Let f(z) = az + b, with ad bc. Where in C is f conformal? cz + d (ii) Let f(z) = z +. What are the images of the real axis, the imaginary axis, and the z unit circle? What are the images of the unit disc and the quadrant {x+iy : x, y > }? (iii) Let C be the unit circle, and C 2 the circle z ( + i) =. Find a Möbius map which simultaneously sends C to the real axis and C 2 to the imaginary axis. The circles divide the plane into four regions. region? Where does your map send each *(iv) Prove that a Möbius map sends the unit disc onto itself if and only if it has the form f(z) = e iθ z + α for some θ R and some α C with α <. ᾱz + 2. Find the images of the following maps, using the principal branches in (ii) and (iv). If you haven t met branches yet then have a go anyway, as though I hadn t mentioned them. (i) f(z) = z 2 on the half-disc {z : z <, Re(z) > } (ii) f(z) = z /3 on the cut plane C \ {x + iy : x, y = } (iii) f(z) = exp z on the half-strip {x + iy : x >, < y < π 2 } (iv) f(z) = log z on the half-disc {z : z <, Re(z) > } 3. For each of the following regions, construct a function which maps it conformally onto the unit disc. If you give a composition of several functions, it would be helpful to provide a sketch to illustrate each step. (i) the open region enclosed between the circles z = and z 2 = 2 (ii) the open quarter-disc {z : z <, arg z (, π 2 )} (iii) the half-strip {x + iy : < x <, y > }. 4. Use the decomposition ( z (z ) 2 = z ) to show that f(z) = z/(z ) 2 is a bijective conformal map from the disc z < to the domain C \ {x + iy : x 4, y = }. *5. Let f be analytic and z C be such that f (z ) = and f (z ). By considering the Taylor expansion of f(z + h) about z, prove that f doubles angles between curves intersecting at z. What happens if f (z ) = f (z ) = and f (z )?

2 6. Consider the map f(z) = 2 (z + z ). Find the points where this map is not conformal and determine how the angles between two curves at those points change at their image. You can quote the results of the previous question here, whether you did it or not. Show that f takes concentric circles with radius r > centered at the origin to cofocal ellipses. What is the image of the unit circle? By considering angles at certain points, sketch the image of the circle z = 2. * Sketch the images of the circles z ( + i) = 5 and z i = 2. Cauchy-Riemann equations ; harmonic functions 7. (i) Let u(x, y) and v(x, y) satisfy the Cauchy-Riemann equations, and define ( z + z g(z, z) = u 2, z z ) ( z + z + iv 2i 2, z z ). 2i Use the chain rule to show that g/ z =. Explain the significance of this result. (ii) Where, if anywhere, in the complex plane are the following functions differentiable, and where are they analytic? Im z ; z 2 ; sech z. (iii) Let f(z) = z 5 / z 4 for z, and f() =. Show that the real and imaginary parts of f satisfy the Cauchy-Riemann equations at z =, but that f is not differentiable there. 8. Let f be analytic on an open set D C. Show that if any of Re f, Im f, f or arg f is constant on D, then f is constant on D. 9. Find complex analytic functions whose real parts are the following: (i) xy (ii) e x cos y (iii) sin x cosh y (iv) y (x + ) 2 + y 2 (v) log(x 2 + y 2 ) You should give your answers as functions of z, rather than of x and y. Where on R 2 are the above functions harmonic, and why?. (i) Let u(x, y) = tan (y/x). Where on R 2 is u defined? Explain why there is no analytic function defined on the whole of C whose real or imaginary part equals u on the domain of u. Show that u is harmonic on {(x, y) : x > } by finding a complex analytic function defined on {x + iy : x > } whose real (or imaginary) part is u. Show similarly that u is harmonic on {(x, y) : x < }. ( ) (ii) Let u(x, y) = tan 2x x 2 + y 2. Where on R 2 is u defined? By considering w(z) = z + i z i and using two cases as in (i), show that u is harmonic where defined. 2

3 . Verify that u(x, y) = e x (x cos y y sin y) is harmonic. Find its harmonic conjugate and determine the family of curves orthogonal to u(x, y) = α, for constant α. Find an analytic function f(z) such that Re f = u. * Can the expression u(z, ) be used to determine f(z) in general? 2. Show that g(z) = exp z maps {x + iy : < y < π} onto {x + iy : y > }. Show that h(z) = sin z maps {x + iy : π 2 < x < π 2, y > } onto {x + iy : y > }. Hence find a conformal map of {x + iy : π 2 < x < π 2, y > } onto {x + iy : < y < π}. Find a function v(x, y) which is harmonic on the strip { π 2 < x < π 2, y > }, with limiting values on the boundaries given by: v = on the two parts of the boundary in the left half-plane, and v = on the two parts of the boundary in the right half-plane. Is your function v unique? You should give v as a function of x and y, rather than of z. *3. Find a function v(x, y) which is harmonic on the unit disc, with limiting values as follows: v = on the part of the boundary in the first quadrant, v = on the part in the third quadrant, and v = on the parts in the second and fourth quadrants. Branches 4. Explain how the principal branch of log z can be used to define a branch of z i which is single-valued on the set D = C \ {x + iy : x, y = }. Does the identity (zw) i = z i w i hold? What is i i for this branch? Using polar coordinates, show that the branch of z i defined above maps D onto an annulus which is covered infinitely often. How would your answers change for a different branch (with the same cut)? 5. Find all branch points of f(z) = [z(z + )] /3. Draw some possible branch cuts in the complex plane. Repeat with f(z) = [z(z + )(z + 2)] /3. * Repeat with f(z) = [z(z + )(z + 2)(z + 3)] /n for n = 2 and n = Let f(z) = (z 2 ) /2. Consider the following two branches of this function. f (z) : branch cut [, ], f (x) = + x 2 for real x > f 2 (z) : branch cut R \ (, ), f 2 (x) = +i x 2 for real x (, ). Find the limiting values of f and f 2 above and below their respective branch cuts. Prove that f is an odd function and f 2 is even, i.e., f ( z) = f (z) and f 2 ( z) = f 2 (z). *7. Let f(z) be a polynomial of even degree. Explain why there is an analytic function g(z), defined on the region z > R for some suitable R, such that g(z) 2 = f(z). When is there such a function defined on the region z < r for some r? 3

4 Series ; singularities 8. Use partial fractions to find the Laurent series of /(z a)(z b)) about z =, where b > a >, in each of the regions z < a, a < z < b and z > b. 9. Find the first two non-zero coefficients in the Taylor series about the origin of the following functions, assuming principal branches for (i), (ii) and (iii). (i) z/ log( + z) ; (ii) cos z ; (iii) log( + e z ) ; (iv) e ez. State the range of values of z for which each series converges. How would your series change for different branches? 2. Find and classify the singularities in the complex plane of the following functions: (i) z 3 (z ) 2 (ii) e z e (z ) 3 (iii) z sinh z (iv) z log z (v) tan z (vi) exp(tan z) (vii) log(tan z) (viii) tan(z ). 2. Find the first three terms of the Laurent series of f(z) = cosec 2 z valid for < z < π. Show that the function g(z) = cosec 2 z z 2 (z + π) 2 (z π) 2, has only removable singularities in z < 2π. Use this to show that, in the Laurent series of f(z) valid for π < z < 2π, the central three non-zero terms are + 3 ( z ) ( π ) π 4 z 2 + * What are the corresponding terms in the series for f(z) valid for nπ < z < (n + )π? Integration Some of the integrals in this section can be evaluated by real methods, but please do them all by contour integral methods. 22. Evaluate C z dz and C z /2 dz (using the principal branch of z /2 ) in the cases: (i) C is the circle z =, and (ii) C is the circle z =. 23. Using Cauchy s theorem or Cauchy s integral formula (but not the residue theorem), evaluate dz in the cases where C is: + z2 C (i) the circle z =, (ii) the circle z i =, and (iii) the circle z = 2. 4

5 24. By evaluating C x2 dz, where C is the ellipse z 2π a 2 cos 2 θ + b 2 sin 2 θ a 2 + y2 = with a, b >, show that b2 dθ = 2π ab. 25. If these results were proved in lectures, feel free to ignore this question. Suppose that at z = z, the function f is non-zero and the function g has a simple zero. Prove that the residue of f/g at z is f(z )/g (z ). Suppose that at z = z, the function f has a pole of order N. Prove that residue of f at z is d N [ ] lim z z (N )! dz N (z z ) N f(z). 26. Evaluate C z 3 e /z + z dz, where C is the circle z = 2, using the following methods. (i) Integrate directly, using the residue theorem. (ii) Apply the substitution w = /z, then use the residue theorem. *(iii) Find the Laurent series valid on C, then integrate term by term. 27. (i) Evaluate dx by closing the contour in the upper half-plane. + x + x2 How does the calculation differ if you close the contour in the lower half-plane? R x (ii) Evaluate lim dx. Why is the limit necessary here? R R + x + x2 (iii) Evaluate e ikx dx for all real k. + x2 28. (i) By integrating around a keyhole contour, show that x a + x dx = π sin πa ( < a < ). Explain why the given restriction on the value of a is necessary. (ii) By integrating around a contour involving the line arg z = 2π n, evaluate dx (n 2). + xn Check by change of variable that your answer agrees with that of part (i). 29. By evaluating each side using a suitable contour integral, show that π sin 2n θ dθ = ( + x 2 dx (n N). ) n+ Find a change of variables that explains why they are equal. 5

6 3. Establish the following: (i) cos x 7π ( + x 2 dx = ) 3 6e ; (ii) sin 2 x x 2 dx = π 2 ; (iii) log x + x 2 dx =. Do (ii) by a contour integral, even though there are other ways see question 37. (log z)2 For (iii), you could integrate around a keyhole contour, or integrate log z around +z 2 +z 2 an arch-shaped contour, i.e. a semicircle with a bump at. (You could do both!) What goes wrong if you integrate log z around a keyhole? +z 2 3. Using a (mostly) rectangular contour involving the line Im z = π, show that Hence, or otherwise, find sin ax sinh x dx = π 2 x sinh x dx. tanh πa 2, where a is real. 32. By considering the integral of n= cot z z 2 + π 2 around a suitable large contour, show that a2 n 2 + a 2 = π coth πa, for ia / Z. a By considering a similar integral, show also that n= (n + a) 2 = π 2 sin 2, for a / Z. πa Deduce from either of these results (you can choose which) the value of n= n 2. *33. By integrating around a contour involving the line arg z = π 4, evaluate cos x 2 dx You may quote that e x2 dx = 2 π. and sin x 2 dx. *34. Let f(z) = z log( + z) with arg( + z) [ π, π), and g(z) = z log z with arg z [ π, π). Explain why h(z) = f(z) g(z) is analytic everywhere outside the unit disc, and justify why h(z) = z log( + /z) there. Find the Laurent series of h(z) about the origin in this region, and hence evaluate h(z) dz. z =2 *35. Evaluate the following using suitable contour integrals, where < a < in (i) and (iii). (i) log(x + ) x a+ dx ; (ii) x 2 + x 2 dx ; (iii) 6 2π log( + a cos θ) dθ.

7 Fourier Transforms 36. Let f(x) = Show directly that { for x < a for x > a f(k) = 2 sin ak k and g(x) = and g(k) = { a x for x < a for x > a 2( cos ak) k 2. Show that g(x) = 2 (f f)(2x), and verify the formula for g(k) using the convolution theorem. Verify by contour integration that the inversion formula holds for f(x). 37. Use the previous question and Parseval s Identity to evaluate. sin 2 x x 2 dx and sin 4 x x 4 dx. 38. Use the Fourier inversion formula (not contour integration) to show that, for a >, e ikx a 2 + k 2 dk = π a e a x and e ikx b 2 + (a + ik) 2 dk = 2π b H(x)e ax sin bx, where H(x) is the Heaviside function. What are the corresponding answers when a <? 39. By considering the convolution of the function f(x) = e x with itself, show that Verify this result by contour integration. e ikx ( + k 2 ) 2 dk = π 2 ( + x )e x. 4. This question shows how the Fourier transform representation of a function reduces to a Fourier series if the function is periodic. Suppose that f(x) has period 2π. Let F (k) = 2π f(x)e ikx dx, and let g(x) = f(x)e a x, where a >. Show that the Fourier transform of g(x) is given by F (k ia) F (k + ia) g(k) = e 2πi(k ia) e 2πi(k+ia). Assuming that F is analytic, sketch the locations of the singularities of g in the complex k-plane. Assuming further that the sequence sup{ g(k) : k = n + 2 } tends to as n, use the Fourier inversion theorem and a suitable contour to show that g(x) = 2π n= for x >, and derive a similar result for x <. F (n)e (in a)x Deduce that f(x) = 2π n= F (n)e inx. 7

8 Laplace Transforms 4. Use standard properties (translation, scaling, etc.) of the Laplace transform to find the Laplace transforms of the following functions: (i) t 3 e 3t, (ii) 2e 3t sin 4t, (iii) e 4t cosh 2t. 42. The function f(t) has Laplace transform F (s) = (i) using partial fractions and standard transforms (ii) using the inversion formula and a contour integral (iii) using standard transforms and the convolution theorem. 43. Find the Laplace transforms of f(t) = t /2 and g(t) = t /2. You may quote that e x2 dx = 2 π. s 3 (s 2. Find f(t) in three ways: + ) Verify, by integrating around a keyhole contour, that the inversion formula holds for f(t). 44. (i) The Gamma function is defined for z C with Re(z) > by Γ(z) = t z e t dt. Show that Γ(z + ) = zγ(z), and deduce that Γ(n + ) = n! if n is a positive integer. Using the previous question, write down the value of Γ( 2 ). For fixed z, find the Laplace transform of f(t) = t z in terms of Γ(z). (ii) The Beta function is defined for z, w C with Re(z), Re(w) > by B(z, w) = t z ( t) w dt. Use the convolution theorem for Laplace transforms to show that B(z, w) = Γ(z)Γ(w) Γ(z + w). (iii) Using question 28, deduce that Γ(z)Γ( z) = For which range of z does this hold? π sin πz. *45. This isn t really to do with Complex Methods it s just a nice use of the results above. Let I m,n = π/2 cos m θ sin n θ dθ, for m, n N. Find I m,n in terms of the Gamma function. Deduce that I m,n is a rational multiple of π if m and n are both even, and is rational otherwise. *46. Using the relation Γ(z + ) = zγ(z), show that we may use analytic continuation to extend the definition of Γ(z) to the whole of C, apart from isolated singularities. Find and classify these singularities and find the residues of Γ(z) at them. Does the relation Γ(z)Γ( z) = π sin πz hold for the continuation? 8

9 Laplace Transforms differential equations In questions 47 5, use Laplace transforms to solve the given equations for t. 47. Solve the differential equation y 3y + 3y y = t 2 e t, with initial conditions y() =, y () =, y () = Solve the differential equation y 3y + 2y = δ(t), with initial conditions y() =, y () =, where δ(t) is the Dirac delta function. For δ(t), take the Laplace transform to be f(t)e st dt, i.e. start just to the left of. ( ) ( ) ( ) ( ) ( ) x 5 x x() Solve the system of differential equations y =, with =. y y() 5. Solve the integral equation f(t) + 4 t Verify that your solution for f(t) is correct. 5. Solve the wave equation with boundary conditions (t τ) f(τ) dτ = t. 2 y t 2 2 y = for x, t, x2 y(, t) = y(x, ) = y t (x, ) = and y(x, t) 2 t2 as x. 52. By considering the Laplace transform of f (t), and assuming that lim t f(t) exists for the second case, prove that f() = lim s sf (s) and lim t f(t) = lim s sf (s). * Show that these still hold if we use the variant Laplace transform given in question 48, providing we replace f() with lim t + f(t). 53. The zeroth-order Bessel function J (t) satisfies the differential equation tj + J + tj =, J () =. ( ) Using the previous question, find the Laplace transform of J (t). Show that J (t) dt =. Find the convolution of J (t) with itself. * By using the inversion formula and a suitable branch cut, show that J (t) = π Verify that this does indeed solve ( ). π cos(t cos θ) dθ. 9