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Chapter 14: Q27P (page 673)
Using the definition (2.1) of , show that the following familiar formulas hold. Hint : Use the same methods as for functions of a real variable.
27..
It is proved that the derivative is,
.
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Describe the Riemann surface for .
Evaluate the integrals by contour integration.
To find that the integrals by computing residue at infinity.
around .
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 in powers of z to get a series convergent for , and in powers of to get a series convergent for .
For each of the following functions w = f(z) = u +iv, find u and v as functions of x and y. Sketch the graph in (x,y) plane of the images of u = const. and v = const. for several values of and several values of as was done for in Figure 9.3. The curves u = const. should be orthogonal to the curves v = const.
w = ez
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