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Schaum's Outline of Complex Variables

379 SOLVED PROBLEMS

Question: 10.10

Prove that sin z = z (1 – z²/π²) (1 – z²/4π²) (1 – z²/9π²) ··· = zΠk=1^∞ (1 – z²/k²π²). ...

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From Problem 7.35, page 233, we have \beg...
Question: 8.4

Prove that the angle between two curves C1 and C2 passing through the point z0 in the z plane [see Figs. 8-1 and 8-2, page 243] is preserved [in magnitude and sense] under the transformation w = f(z), i.e., the mapping is conformal, if f(z) is analytic at z0 and f′0(z0)≠0. ...

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By Problem 8.3, each curve is rotated through the ...
Question: 10.47

Show that for large positive values of z, Jn(z) ∼ √2/πz cos (z – nπ/2 – π/4) ...

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By Problem 6.33, we have J_{n}(z)=\frac{1}{...
Question: 10.46

Prove that for m = 1, 2, 3, … , Γ(1/m)Γ(2/m)Γ(3/m) ··· Γ(m – 1/m) = (2π)^(m-1)/2 /√m ...

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We have P=\Gamma\left(\frac{1}{m}\right) \G...
Question: 7.36

Evaluate 1/2πi ∫a-i∞^a+i∞ e^zt/√z + 1 dz where a and t are any positive constants. ...

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The integrand has a branch point at z=-1[/l...
Question: 10.45

Prove that Pn(z) = F (-n, n + 1; 1; 1 – z/2), n = 0, 1, 2, 3, … . ...

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The Legendre polynomials P_{n}(z) a...
Question: 7.35

Prove that cot z = 1/z + 2z{1/z² – π2 + 1/z² – 4π² + ···}. ...

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We can write the result of Problem 7.34 in the for...
Question: 7.34

Prove that cot z = 1/z + ∑n(1/z – nπ + 1/nπ) where the summation extends over n = ±1, ±2, . . . . ...

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Consider the function f(z)=\cot z-\frac{1}{...
Question: 7.33

Prove Mittag –Leffler’s expansion theorem (see page 209). ...

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Let f(z) have poles at z=a_{...
Question: 10.43

Prove that (a) sn(z + 2iK′) = sn z, (b) cn(z + 2K + 2iK′) = cn z, (c) dn(z + 4iK′) = dn z. ...

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Using Problems 10.39, 10.42, 10.170, and the addit...
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