Problem sheet 8 March 1st 2005

MT290 Complex variable

Ex. 1 Give the radius of convergence for each of the following series:

a. ∑_{n=1}^{∞} \frac{z^{n}}{n^{3}} ; b. ∑_{n=0}^{∞} \frac{z^{n}}{5^{n}}; c. ∑_{n=0}^{∞} \frac{z^{n}}{n^{3} + 5^{n}}; d. ∑_{n=0}^{∞} z^{n}5^{−n^{2}}

Ex. 2 Determine the radius of convergence for:

a) ∑_{n=0}^{∞} \frac{z^{n}}{7^{n}} ,  b) ∑_{n=0}^{∞} \frac{z^{5n}}{7^{n}}  c) ∑_{n=0}^{∞} \frac{z^{bn}}{a^{n}}  d) ∑_{n=0}^{∞} \frac{z^{n}}{n^{r}}  e) ∑_{n=0}^{∞} \frac{z^{n}}{\binom{3n}{n}}.

where in c) a and b and in d) r are positive real constants. Recall that \binom{m}{n} = \frac{m!}{n!(m−n)!} and use Stirling’s approximation for n! below.

Ex. 3 Determine the integrals, if they exist. Distinguish the various radii R.

a)  ∫_{|z|=R} \frac{dz}{z(z + 3)},  b)  ∫_{|z|=R} \frac{sin(πz)dz}{z(2z − 1)(z − 2)}.

Ex. 4 (Revision) The function f(z) = u(x, y) + iv(x, y) is differentiable for all z = x + iy, and f(0) = 0. Given that u(x, y) = sin x cosh y, show, using the Cauchy-Riemann equations, that f(z) = sin z. Hint: You will need to use results like sin(iy) = i sinh y and remember trigonometric identities like sin(A + B) = . . ..

Ex. 5 Stirling’s formula is: n! ∼ \sqrt{2πn} (\frac{n}{e})^{n} . Use your pocket calculator to convince yourself for n = 10, 20, 30, .... List some values of \frac{n!}{\sqrt{2πn}(\frac{n}{e})^{n}}

Use this to find an approximation for \binom{2n}{n} and \binom{3n}{n}. Try to prove a much weaker form of Stirling’s formula as follows: ln(n!) = ln 1 + ln 2 + … + ln n. Approximate the sum by a definite integral like ∫_{a}^{b} ln x dx with appropriate bounds. Find a) a lower and b) an upper bound for n!. If these bounds are not too far apart you have a good approximation to n!.

Ex. 6 (More difficult) (This problem shows that a) not only semicircle contours help to evaluate integrals along the real line; b) that you can reduce an integral like ∫ e^{−x^{2}} cos ax dx (here with a = 1) to ∫ e^{−x^{2}}, dx. The latter is not really easy, but at least well known so that you can use it here.)

Let f : ℂ → ℂ be the function with f(z) = e^{−z^{2}}. Let R(K) denote the rectangle defined by the four points P_{1} = −K + 0i, P_{2} = +K + 0i, P_{3} = +K + \frac{1}{2}i, P_{4} = −K + \frac{1}{2}i. Let γ_{1} denote the path along the edge connecting P_{1} and P_{2}, Let γ_{2} denote the path along the edge connecting P_{2} and P_{3}, Let γ_{3} denote the path along the edge connecting P_{3} and P_{4}, Let γ_{4} denote the path along the edge connecting P_{4} and P_{1}.

i) Draw the integration contour in the Argand diagram.

ii) Show that ∫_{∂R(K)} f(z) dz = 0. Here ∂R(K) denotes the boundary of the rectangle, taken in anticlockwise direction.

iii) Show that lim_{K→∞} ∫_{γ_{2}} f(z) dz = 0, and similarly lim_{K→∞} ∫_{γ_{4}} f(z) dz = 0. Use the above results, and (without

proof) the well known result ∫_{−∞}^{∞} e^{−x^{2}} dx = \sqrt{π} to conclude that ∫_{0}^{∞} e^{−x^{2}} cos x dx = \frac{\sqrt{π}}{2e^{1/4}} .