Chapter 14

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer. $$\frac{z}{1-z^4}\text{ at }z=i$$

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer. $$\frac{z^2}{z^4+16}\text{ at }z=\sqrt{2}(1+i)$$

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer. $$\frac{e^{2z}}{1+e^z}\text{ at }z=i\pi$$

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer. $$\frac{e^{iz}}{9z^2+4}\text{ at }z=\frac{2i}{3}$$

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer. $$\frac{1-\cos 2z}{z^3}\text{ at }z=0$$

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer. $$\frac{e^{2\pi iz}}{1-z^3}\text{ at }z=e^{2\pi i/3}$$

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer. $$\frac{z+2}{(z^2+9)(z^2+1)}\text{ at }z=3i$$

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer. $$\frac{e^{2z}}{4\cosh z-5}\text{ at }z=\ln 2$$

Find the residues of the following functions at the indicated points. Try to select the easiest of the methods outlined above. Check your results by computer. $$\frac{e^{3z}-3z-1}{z^4}\text{ at }z=0$$

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 $(d/dz)\left(e^z\right)=e^z$.