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Chapter 14: Q29P (page 687)
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.
at
The residue of the function at is
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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.
Use the following sequence of mappings to find the steady state temperature in the semi-infinite strip if and as . (See Chapter 13, Section 2 and Problem 2.6.)
Useto map the half plane on the upper half plane , with the positive axis corresponding to the two rays and , and the negative yaxis corresponding to the interval of the x'axis. Use z'=-coszto map the half-stripon the Z'half plane described in (a). The interval role="math" localid="1664365839099" corresponds to the baseof the strip.
Comments: The temperature problem in the (u,v) plane is like the problems shown in the z plane of Figures 10.1 and 10.2, and so is given by . In the z plane you will find
Put and use the formula for to get " width="9" height="19" role="math">
Note that this is the same answer as in Chapter 13 Problem 2.6, if we replace 10 by .
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
To find: uand v as a function of x and y & plot the graph and show curve u = constant constant should be orthogonal to the curves v = constant . w = sin z
Let f(z) be expanded in the Laurent series that is valid for all z outside some circle, that is,(see Section 4). This series is called the Laurent series "about infinity." Show that the result of integrating the Laurent series term by term around a very large circle (of radius > M) in the positive direction, is (just as in the original proof of the residue theorem in Section 5). Remember that the integral "around " is taken in the negative direction, and is equal to : (residue at ). Conclude that . Caution: In using this method of computing be sure you have the Laurent series that converges for all sufficiently large z.
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