Problem sheet 10 March 15th 2005

MT290 Complex variable

Ex. 1 Find the order of the zero for each of the following functions at z = 0:

a. z^{100} − z^{10};  b. e^{z} − 1 − z;  c. \frac{sin z − z}{z};  d.\frac{cosh z − 1 − \frac{z^{2}}{2}}{z^{2}} .

Ex. 2 Find the singularities of the following functions and determine whether or not they are poles:

\frac{z^{2}}{(z − 2)^{5}(z − 5)^{7}(z − 10)^{100}} ;  \frac{e^{z}}{z^{2}} − (\frac{1}{z} + \frac{1}{2})^{2} ;  cosec(1/z).

Where the functions have poles, determine the order of the poles. Compare sin \frac{1}{z} and \frac{1}{sin z}.

Ex. 3 Find the poles (with their order), and residues at the poles, for the following functions: a) \frac{z}{z^{2} − 3z + 2} b)

\frac{1}{e^{z} + 1}  c) \frac{sin z}{z^{3}}  d) \frac{e^{z^{2}}}{(z^{2} + 9)(z^{2} + 25)}. Hint The function in b) has infinitely many poles.

Ex. 4 Find the poles of the following functions which have positive imaginary part, and calculate the residues at the poles.

\frac{z^{2}}{(z^{2} + 1)(z^{2} + 4)}; \frac{e^{iz}}{(z^{2} + 1)(z^{2} + 4)}.

If you have any questions on the course, previous problem sheets, exam or whatever, please let me know. In the last week I might come back to questions of general interest in the lecture.