Mathematical Methods in the Physical Sciences · Problem 2.6

FUNCTIONS OF A COMPLEX VARIABLE — Mary Boas

Mathematics — Methods · laplace-equation-series-summation

Problem

Show that the series in (2.12) can be summed to get

T = \frac{200}{\pi} \arctan\left(\frac{\sin(\pi x/10)}{\sinh(\pi y/10)}\right)

(with the arc tangent in radians). Use this formula to check the value

T = 26.1^\circ

x = y = 5

Hints

for summing the series: Use

\sin(n\pi x/10) = \mathrm{Im}\, e^{in\pi x/10}

to write the series as

\mathrm{Im}\sum_{\mathrm{odd}\,n} z^n/n

. (What is

z

?) Compare this with the series for

\ln[(1+z)/(1-z)]

(see Chapter 1, Problem 13.22). Then use (13.5) of Chapter 2.