Solving for z: 2Pz = e^(zL) - e^(-zL)

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Homework Statement


Solve for z: 2Pz = e^(zL) - e^(-zL)


Homework Equations





The Attempt at a Solution

 
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Generally speaking, equations that involve the unknown variable both inside and outside a transcendental function cannot be solved in terms of elementary fuctions.

They can sometimes be solved in terms of "special functions". Here it looks like you ought to be able to solve that using "Lambert's W function" which is defined as the inverse function to f(x)= xex: W(xe^x)= x.
 

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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