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Chapter 14: Q28P (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.
28.. (See hint below.)
Problem 28 is the chin rule for the derivative of a function of a function.
It is proved that the derivative is,
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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.
Show that equation (4.4) can be written as (4.5). Then expand each of the fractions in the parenthesis in (4.5) in powers of z and in powers of [see equation (4.7) ] and combine the series to obtain (4.6), (4.8), and (4.2). 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 .
We have discussed the fact that a conformal transformation magnifies and rotates an infinitesimal geometrical figure. We showed that is the magnification factor. Show that the angle of is the rotation angle. Hint: Consider the rotation and magnification of an arc (of length and angle arctan which is required to obtain the image of dz , namely dw.
Describe the Riemann surface for .
Find the inverse Laplace transform of the following functions by using (7.16).
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