I have forgotten my complex calculus

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Can anyone help me with a complex calculus concept? I took this class years ago and cannot remember how to simplify something like \left|cos(1+2i)\right|^2?

Thanks!

Michael
 
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If a is a complex number, |a|2 = a * conjugate(a)

where if a=x+iy, conjugate(a) = x-iy.

So re-write cos(1+2i) in terms of exponentials and multiply... something good probably comes out the other side
 
Thanks! I'll try that. I think I actually remember now.

Michael
 

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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