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Chapter 14: Q26P (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.
26..
It is proved that the derivative is,
.
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A fluid flow is called irrotational if 鈭嚸梀 = 0 where V = velocity of fluid (Chapter 6, Section 11); then V = 鈭囄�. Use Problem 10.15 of Chapter 6 to show that if the fluid is incompressible, the 桅 satisfies Laplace鈥檚 equation. (Caution: In Chapter 6, we used V = v蟻, with v = velocity; here V = velocity.)
Evaluate the following integrals by computing residues at infinity. Check your answers by computing residues at all the finite poles. (It is understood that means in the positive direction.)
around
Find the inverse Laplace transform of the following functions using (7.16) .
w =鈭歾. Hint: This is equivalent to w2 = z; find x and y in terms of u and v and then solve the pair of equations for u and v in terms of x and y. Note that this is really the same problem as Problem 1 with the z and w planes interchanged.
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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