Problem sheet 9 March 8th 2005

MT290 Complex variable

Ex. 1 Find power series for the following functions about the points stated and give the radius of convergence for each of the series. a. \frac{1}{2 − z} about z = 0; b. \frac{1}{2 − z} about z = 12; c. \frac{5}{(1 − z)(4 + z)} about z = 0. d. e^{z} about z = i. e. \frac{1}{3 − z}

about z = 4i.

Ex. 2 Determine the power series of sin z in two different ways:

a) Use the definition of sin in terms of the complex exp function.

b) Use \frac{d^{2} sin z}{dz^{2}} = − sin z, cos(0) = 1 and the fact that sin is an odd function.

c) Use the definition of sinh z in terms of the exp to find its power series. Compare with sin z and deduce that sin(iz) = i sinh z.

Can you simplify the above calculations by means of Taylor’s theorem?

Ex. 3 Assuming that it is alright to integrate a power series term by term within its radius of convergence (it is !) use the series for (1 + z)^{−1} to obtain the power series:

log(1 + z) = ∑_{n=1}^{∞} \frac{(−1)^{n+1}}{n} z^{n}.  (∗)

What is the radius of convergence of this series? Let z = iy in (*). Take the imaginary part to obtain the series for arctan y.

Ex. 4 Find the first few coefficients of the Taylor series of tan z in two different ways.

a) using c_{n} = \frac{f^{(n)}(0)}{n!}. When you are tired of differentiating try

b)

tan z = \frac{sin z}{cos z} = \frac{z − \frac{z^{3}}{3!} + \frac{z^{5}}{5!} ∓}{1 − \frac{z^{2}}{2!} + \frac{z^{4}}{4!} ∓} = c_{0} + c_{1}z + c_{2}z^{2} + …

Multiply by cos z and find c_{0}, c_{2}, c_{4}, … Then find, c_{1}, from this c_{3} etc.

Ex. 5 Complete the following explanation of Taylor’s theorem: Perhaps you wondered where this formula c_{n} = \frac{f^{(n)}(0)}{n!} comes from. Let’s try to filter out the coefficient c_{k} from f(z) = ∑_{n=0}^{∞} c_{n}z^{n}. A good filter is our favourite integral

∫_{|z|=1} z^{n} dz = \begin{cases}2πi & if n = −1 \\ 0 & otherwise. \end{cases}

∫_{|z|=1} f(z) dz would not lead anywhere, for a differentiable function this is just 0. But let’s try

∫_{|z|=1} \frac{f(z)}{z^{k+1}} dz = ∫_{|z|=1} \frac{∑_{n=0}^{∞} c_{n}z^{n}}{z^{k+1}} dz = ∫_{|z|=1} ∑_{n=0}^{∞} c_{n} \frac{z^{n}}{z^{k+1}} dz

Now go on, exchange the integration and summation (you are allowed to do it!), use our filter, combine with Cauchy’s integral formulae and find the expression for c_{k}.

Note: Reminder/Workshops: Thursday 11am: 325 Thursday 3pm: 325 But no Friday workshops any longer. If you have questions on the course, home work solutions or feel unhappy about certain topics: (in addition to the workshops, office hour...) I might have time to address these during the last week in the lecture. General questions for this are very welcome! Otherwise I will possibly explain some selected old exam problems.