Applied Complex Variables by John W. Dettman

Author:John W. Dettman
Language: eng
Format: epub
Publisher: Dover Publications, Inc.
Published: 1965-03-14T16:00:00+00:00

To show that

we have to show that . On the horizontal sides of the square Cn, ζ = ξ ± nπi, and |tan ζ| ≤ |coth nπ|, which is bounded for n = 1, 2, 3, . . . . Hence, on the horizontal sides

as n → ∞. On the vertical sides ζ = ±nπ + iη, and |tan ζ| ≤ |tanh η| ≤ tanh nπ which is bounded for n = 1, 2, 3, . . . . Hence, on the vertical sides

as n → ∞. Evidently the convergence is normal in the unextended plane excluding the poles of tan z.

EXAMPLE 5.4.1. Find an expansion for cot z. This time the function has a simple pole at the origin and if we proceed as before we shall have to compute the residue at a double pole of the integrand at the origin. It is more convenient to subtract out the principal part of cot z at the origin before we expand. Therefore, we look for an expansion of cot z – 1/z. Consider the integral

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