How to Find the Conjugate of a Complex Exponential Function?

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Homework Statement



Find the conjugate of

\varphi=exp(-x^2/x_0^2)

Homework Equations





The Attempt at a Solution




Isn't the conjugate \varphi*=exp(x^2/x_0^2)
 
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noblegas said:

Homework Statement



Find the conjugate of

\varphi=exp(-x^2/x_0^2)

Homework Equations


The Attempt at a Solution

Isn't the conjugate \varphi*=exp(x^2/x_0^2)

Not if x and x0 are real, which I suspect they are. What is it in that case?
 
Last edited:
Dick said:
Not if x and x0 are real, which I suspect they are. What is it in that case?

oh ,my solution would only be correct if x/x0 is imaginary.would my expression
<br /> exp(-x^2/x_0^2)<br /> not change when taking its conjugate??
 
noblegas said:
oh ,my solution would only be correct if x/x0 is imaginary.would my expression
<br /> exp(-x^2/x_0^2)<br /> not change when taking its conjugate??

Right, sort of. If x is imaginary the conjugate(exp(x))=exp(-x). If x is real then conjugate(exp(x))=exp(x). But your solution is only correct if (x/x0)^2 is purely imaginary.
 
Dick said:
Right, sort of. If x is imaginary the conjugate(exp(x))=exp(-x). If x is real then conjugate(exp(x))=exp(x). But your solution is only correct if (x/x0)^2 is purely imaginary.

but x/x0 is not purely imaginary , but completely real. So my expression would remain the same when taking its conjugate
 
noblegas said:
but x/x0 is not purely imaginary , but completely real. So my expression would remain the same when taking its conjugate

Yesss.
 

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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