Complex Analysis: Theory and Applications · Problem 4.3.11

Elementary Functions — Howell & Matthews

Mathematics — Complex Analysis · power-series-functional-equation

Problem

For ∣z∣<1|z| < 1, let f(z)=∑n=0∞z(2n)=z+z2+z4+⋯+z2n+⋯f(z) = \displaystyle\sum_{n=0}^{\infty} z^{(2^n)} = z + z^2 + z^4 + \cdots + z^{2^n} + \cdots. Show that f(z)=z+f(z2)f(z) = z + f(z^2).

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