Complex Methods: Example Sheet 1

Transcription

1 Complex Methods: Example Sheet Part IB, Lent Term 26 Dr R. E. Hunt Cauch Riemann equations. (i) Where, if anwhere, in the complex plane are the following functions differentiable, and where are the analtic? Imz; z 2 ; sechz. Copright 26 Universit of Cambridge. Not to be quoted or reproduced without permission. (ii) Let f(z) = z 5 / z 4, z, f() =. Show that the real and imaginar parts of f satisf the Cauch Riemann equations at z =, but that f is not differentiable atz =. 2. Find, as functions ofz, complex analtic functions f(z) whose real parts are the following: (i) x (ii) x (iii) sin x cosh (iv) log(x ) (v) (x+) (vi) tan ( 2x x 2 2 Deduce that the above functions are harmonic on appropriatel-chosen domains, which ou should specif. 3. B consideringw(z) = (i+z)/(i z), show that φ(x,) = tan 2x x is harmonic. 4. Verif that the functionφ(x,) = e x (xcos sin) is harmonic. Find its harmonic conjugate and, b considering φ or otherwise, determine the famil of curves orthogonal to φ(x,) = c for a given constant c. Find an analtic function f(z) such that Ref = φ. Can the expression f(z) = φ(z,) be used to determinef(z) in general? Branches of multi-valued functions 5. Show how the principal branch of logz can be used to define a branch of z i which is singlevalued on the domain D = C\R. Evaluatei i for this branch. Show, using polar coordinates, that the branch of z i defined above maps D onto an annulus which is covered infinitel often. How would our answers change, if at all, for a different branch? 6. How man branch points does f(z) = [z(z +)] /3 have? Draw some possible branch cuts in the complex plane and on the Riemann sphere. Repeat forf(z) = (z 2 +) /2. Repeat also forf(z) = [z(z +)(z +2)] /3 and f(z) = [z(z +)(z +2)(z +3)] /2. ) 7. Letf(z) = (z 2 ) /2, and consider two different branches of the functionf(z): f (z) : branch cut[,], f (x) = + x 2 for realx > ; f 2 (z) : branch cut(, ] [, ), 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, i.e.,f (z) = f ( z), and that f 2 is even.

2 Conformal mappings 8. How does the disc z < transform under the mapping z z? Use the identit ( z (z ) 2 = z ) to show that the map f(z) = z/(z ) 2 is a one-to-one conformal mapping of the disc z < onto the domainc\{x+i : x 4, = }. Copright 26 Universit of Cambridge. Not to be quoted or reproduced without permission. 9. Find conformal mappings f i of U i onto V i for each of the following cases. If the mapping is a composition of several functions, provide a sketch for each step. D denotes the unit disc {z : z < }. (i) U = {z : Rez <, < Imz < }, V = D. (ii) U 2 = D, V 2 is the cut complex plane C\R. (iii) U 3 is the angular sector{z : < argz < α}, V 3 = {z : < Imz < }. (iv) U 4 is the open region bounded between two circles { z : z <, z +i > 2 }, V 4 = D. Laplace s equation. Show that g(z) = e z maps the strip S = {z : < Imz < π} onto the UHP {z : Imz > }, h(z) = sinz maps the half-strip H = {z : π 2 < Rez < π 2, Imz > } onto the UHP. Find a conformal map f: H S. Hence find a function φ(x,) which is harmonic on the half-strip H with the following limiting values on its boundar H : { on H in the LHP (x < ), φ(x,) = on H in the RHP (x > ). Giveφas a function ofxand. Is there onl one such function?. Using conformal mapping(s), find a solution to Laplace s equation in the upper half-plane {z : Im z > } with boundar conditions { x [,], φ(x,) = otherwise. [Find a mapf of the upper half-plane onto itself that makes the boundar conditions easier to deal with.] Comments on or corrections to this problem sheet are ver welcome and ma be sent to reh@cam.ac.uk.

3 Talor and Laurent series Complex Methods: Example Sheet 2 Part IB, Lent Term 26 Dr R. E. Hunt. Find the first two non-vanishing coefficients in the Talor expansion about the origin of the following functions, assuming principal branches when there is a choice. You ma make use of standard series expansions forlog(+z), etc. (i) z/log(+z) (ii) (cosz) /2 (iii) log(+e z ) (iv) e ez. Copright 26 Universit of Cambridge. Not to be quoted or reproduced without permission. State the range of values ofz for which each series converges. How would our answers differ if ou assumed branches different from the principal branch? 2. Let a, b be complex constants with < a < b. Use partial fractions to find the Laurent expansions of /{(z a)(z b)} about z = in each of the regions z < a, a < z < b and z > b. 3. Find the first three terms of the Laurent expansion of f(z) = cosec 2 z valid for < z < π. Show that the functionh(z) = f(z) z 2 (z+π) 2 (z π) 2 has onl removable singularities in z < 2π. Explain how to remove them to obtain a functionh(z) analtic in that region. Find a Talor series for H(z) about the origin and explain wh it must be convergent in z < 2π. Hence, or otherwise, find the three non-zero central terms of the Laurent expansion of f(z) valid forπ < z < 2π. 4. Write down the location and tpe of each of the singularities of the following functions: e z e (i) z 3 (z ) 2 (ii) tanz (iii) zcothz (iv) ( z) 3 z (v) exp(tan z) (vi) sinh z 2 (vii) log(+e z ) (viii) tan(z ). Integration and residues 5. Evaluate zdz whenγ is the circle z =, and when γ is the circle z =. γ 6. (i) Show that iff(z) is analtic, then the residue off(z)/(z z ) atz = z is f(z ). (ii) Show that if/f(z) has a simple pole atz = z, then its residue atz = z is/f (z ). (iii) Show that ifh(z) has a simple zero atz = z andg(z) is analtic and non-zero, the residue of g(z)/h(z) at z = z is g(z )/h (z ). (iv) Prove the formula for the residue of a functionf(z) that has a pole of ordern atz = z : { d N ( lim (z z z z (N )! dz N ) N f(z) ) }. 7. Evaluate, using Cauch s theorem or the residue theorem, dz dz e z coszdz (i) +z 2 (ii) +z 2 (iii) (+z 2 )sinz γ γ 2 γ 3 (iv) z 3 e /z dz γ 4 +z whereγ is the ellipsex =,γ 2 is the circlex = 2,γ 3 is the circle z (2+i) = 2 and γ 4 is the circle z = 2.

4 8. B integrating the functionz n (z a) (z a ) around the unit circle in thez-plane (where a is real,a >, and n is a non-negative integer), evaluate 2π cosnθ 2acosθ +a 2 dθ. The calculus of residues 9. Evaluate lim R R R xdx +x+x 2. Wh is the limit here rather than just the integral?. B integrating around a kehole contour, show that Copright 26 Universit of Cambridge. Not to be quoted or reproduced without permission. x a dx +x = π sin(πa) ( < a < ). Explain wh the given restrictions on the value ofaare necessar.. B integrating around a contour involving the real axis and the line z = re 2πi/n, evaluate dx +x n (n 2). Check (b change of variable) that our answer agrees with that of the previous question. 2. Establish the following: (i) cosx 7π (+x 2 dx = ) 3 6e (ii) sin 2 x x 2 dx = π 2 (iii) logx dx =. +x2 [For part (iii), integrate (logz) 2 /( + z 2 ) around a kehole, or logz/( + z 2 ) along the real axis (or both). What goes wrong if ou integratelogz/(+z 2 ) around a kehole?] 3. LetP(z) be a non-constant polnomial. Consider the contour integral P (z) I = P(z) dz. Show that, if γ is a contour that encloses no zeros of P, then I =. γ Evaluate the limit of I as R, where γ is the circle z = R, and deduce that P has at least one zero in the complex plane. 4. B considering the integral of f(z) = cotz/(z 2 +π 2 a 2 ) around a suitable large contour, prove that, providedia is not an integer, n 2 +a 2 = π a coth(πa). B considering a similar integral prove also that, ifais not an integer, Find an expression for n= (n+a) 2 = π 2 sin 2 (πa). n 2 +a 2 and take the limit as a to deduce the value of n 2. n= Comments on or corrections to this problem sheet are ver welcome and ma be sent to reh@cam.ac.uk.

5 Complex Methods: Example Sheet 3 Part IB, Lent Term 26 Dr R. E. Hunt Fourier Transforms. Let x < 2 f(x) = a, otherwise; a x x < a, g(x) = otherwise. Copright 26 Universit of Cambridge. Not to be quoted or reproduced without permission. Show that f(k) = 2 k sin ak 2 and g(k) = 4 k 2 sin2 ak 2. Verif b contour integration the inversion formula for f(x), including the value at x = 2 a. What is the convolution off with itself? 2. Using the results of the previous question and Parseval s identit, evaluate the integrals sin 2 x x 2 dx and sin 4 x x 4 dx. 3. B using the relationship between the Fourier transform and its inverse, show that for real a and b with a >, e a t = a π ω 2 +a 2 eiωt dω and e at sinbth(t) = b 2π (iω +a) 2 +b 2 eiωt dω where H(t) is the Heaviside step function. What are the values of the integrals when a <? What happens when a =? 4. Show that the convolution of the functione x with itself is given bf(x) = (+ x )e x. Use the convolution theorem to show that f(x) = 2 π and verif this result b contour integration. e ikx (+k 2 ) 2 dk 5. Suppose that f(x) has period 2π and let g(x) = f(x)e a x where a >. Show that the Fourier Transform ofg is given b g(k) = F(k ia) F(k +ia) e 2πi(k ia) e 2πi(k+ia) where F(k) = 2π f(x)e ikx dx. Assuming that F is analtic, sketch the locations of the singularities of g in the complex k-plane. Further assuming that F decas sufficientl quickl at infinit, use the Fourier inversion theorem and a suitable contour to show that g(x) = 2π forx > and derive a similar result when x <. F(n)e (in a)x

6 Deduce that f(x) = 2π F(n)e inx. [This shows how the Fourier transform representation of a function reduces to a Fourier series if the function is periodic.] Laplace Transforms 6. Starting from the Laplace transform of (namel p ), and using onl standard properties of the Laplace transform (shifting, etc.), find the Laplace transforms of the following functions: (i)e 2t (ii)t 3 e 3t (iii)e 3t sin4t (iv)e 4t cosh2t. Copright 26 Universit of Cambridge. Not to be quoted or reproduced without permission. 7. Using partial fractions and expressions for the Laplace transforms of elementar functions, find the inverse Laplace transform of f(p) = (p+3)/{(p 2)(p 2 +)}. Verif this result using the Bromwich inversion formula. 8. Use Laplace transforms to solve the differential equation 3 +3 = t 2 e t with initial conditions() =, () =, () = Solve the integral equationf(t)+4 solution. t. Solve the sstem of differential equations d dt (t τ)f(τ)dτ = t for the unknown functionf. Verif our ( ) x = ( 5 )( x ) with. The zeroth order Bessel function J (t) satisfies the differential equation tj +J +tj = ( ) x = ( ) 3 att =. withj () = (andj () = from the equation). Find the Laplace transform ofj and deduce that J (t)dt =. Find the convolution of J with itself. 2. Use Laplace transforms to solve the heat equation T/ t = 2 T/ x 2 with boundar conditions T(x,) = 3sin2πx ( < x < ),T(,t) = T(,t) = (t > ). 3. Using the equalit e x2 dx = 2 π, find the Laplace transform of f(t) = t /2. B integrating around a Bromwich kehole contour, verif the inversion formula for f(t). What is the Laplace transform oft /2? 4. The gamma and beta functions are defined forz,w C b Γ(z) = t z e t dt and B(z,w) = t z ( t) w dt whenrez,rew > (and b analtic continuation elsewhere). Show thatγ(z+) = zγ(z) and hence thatγ(n+) = n! ifnis a positive integer. Using the previous question, write down the value of Γ( 2 ). For a fixed value of z, find the Laplace transform of f(t) = t z in terms of Γ(z). Find the Laplace transform of the convolutiont z t w. Hence establish that B(z,w) = Γ(z)Γ(w) Γ(z +w).