91Ó°ÊÓ

For each of the following functions find the first few terms of each of the Laurent series about the origin, that is, one series for each annular ring between singular points. Find the residue of each function at the origin. ( Warning: To find the residue, you must use the Laurent series which converges near the origin.) Hints: See Problem 2. Use partial fractions as in equations (4.5) and \((4.7) .\) Expand a term \(1 /(z-a)\) in powers of \(z\) to get a series convergent for \(|z|a\) \(\frac{z-1}{z^{3}(z-2)}\)

Using the definition of \(e^{z}\) by its power series \([(8.1) \text { of Chapter } 2],\) and the theorem (Chapters 1 and 2) that power series may be differentiated term by term (within the disk of convergence), and the result of Problem \(30,\) show that \((d / d z)\left(e^{z}\right)=e^{z}\).

Using series you know from Chapter 1, write the power series (about the origin) of the following functions. Use Theorem III to find the disk of convergence of each series. What you are looking for is the point (anywhere in the complex plane) nearest the origin, at which the function does not have a derivative. Then the disk of convergence has center at the origin and extends to that point. The series converges inside the disk. $$\sqrt{1+z^{2}}$$

Show that the following functions are harmonic, that is, that they satisfy Laplace's equation, and find for each a function \(f(z)\) of which the given function is the real part. Show that the function \(v(x, y)\) (which you find) also satisfies Laplace's equation. $$\cosh y \cos x$$