Complex variables, maximum value

babyrudin · 2008-03-19T01:57:42+00:00

Homework Equations Let w \in C be a fixed complex number with |w|<1. Let f(z)=\frac{z-w}{1-\bar{w}z}. Calculate its maximum value in the region |z| \leq 1. The Attempt at a Solution How should I approach this? Not sure if maximum modulus principle is of much help.

Thread 'Prove that the integral is equal to ##\pi^2/8##'

Prove $$\int\limits_0^{\sqrt2/4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx = \frac{\pi^2}{8}.$$ Let $$I = \int\limits_0^{\sqrt 2 / 4}\frac{1}{\sqrt{x-x^2}}\arcsin\sqrt{\frac{(x-1)\left(x-1+x\sqrt{9-16x}\right)}{1-2x}} \, \mathrm dx. \tag{1}$$ The representation integral of ##\arcsin## is $$\arcsin u = \int\limits_{0}^{1} \frac{\mathrm dt}{\sqrt{1-t^2}}, \qquad 0 \leqslant u \leqslant 1.$$ Plugging identity above into ##(1)## with ##u...

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Homework Equations

The Attempt at a Solution

Thread 'Prove that the integral is equal to ##\pi^2/8##'

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