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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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Use the Cauchy-Riemann conditions to find out whether the functions in Problems 1.1 to 1.21 are analytic.
A fluid flow is called irrotational if ∇×V = 0 where V = velocity of fluid (Chapter 6, Section 11); then V = ∇Φ. Use Problem 10.15 of Chapter 6 to show that if the fluid is incompressible, the Φ satisfies Laplace’s equation. (Caution: In Chapter 6, we used V = vρ, with v = velocity; here V = velocity.)
Evaluate the integrals by contour integration.
Find the real and imaginary parts and of the following functions.
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.
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