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Unformatted text preview: nic. A pair u, v of harmonic functions that also satisfy CR are called conjugate harmonic functions.
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
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
(b) f: C ÆC given by f(z) = z– is differentiable nowhere.
(c) f: C ÆC given by f(z) = |z| is differentiable nowhere except possibly at zero.
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
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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