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Ii 2 c f c c given by fz z is differentiable nowhere

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Unformatted text preview: nic. A pair u, v of harmonic functions that also satisfy CR are called conjugate harmonic functions. 8 Example 3 2 Let u(x, y) = x - 3xy - 5y. Show that u is harmonic, and find a conjugate for it. Example 2.9 Write f(z) = 1/z in this form. Exercise Set 2 p. 806, #1, 5, 9, 15, 19, 21, 23, 25, 31, 35 p. 810 #1, 5, 9, 15, 25, 32 Hand In 1. Using the fact (shown in class) that f(z) = Re(z) is differentiable nowhere, and the formal rules for differentiation but not the C-R condition, deduce each of the following: (a) f: C ÆC given by f(z) = Im(z) is differentiable nowhere II (b) f: C ÆC given by f(z) = z– is differentiable nowhere. II 2 (c) f: C ÆC given by f(z) = |z| is differentiable nowhere except possibly at zero. II 2 2. Now show that f(z) = |z| is, in fact, differentiable at z = 0. 3. Transcendental Functions Definition 3.1. The exponential complex function exp: C ÆC is given by II x exp(z) = e (cos y + i sin y), for z = x+iy. This is also written as ez, for reasons we saw in the last section....
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