Division of the complex plane into congruent rectangles

Reference no: EM131350757

Suppose τ is purely imaginary, say τ = it with t > 0.

Consider the division of the complex plane into congruent rectangles obtained by considering the lines x = n/2, y = tm/2 as n and m range over the integers. (An example is the rectangle whose vertices are 0, 1/2, 1/2 + τ /2, and τ /2.)

(a) Show that ℘ is real-valued on all these lines, and hence on the boundaries of all these rectangles.

(b) Prove that ℘ maps the interior of each rectangle conformally to the upper (or lower) half-plane.

Reference no: EM131350757

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