Integral of a complex conjugate: why int_{z*} = (z^2)/2 instead of ((z*)^2)/2 ?

[Mondo]

Hello,
I have a simple question, why [imath]\int_{z^*} = \frac{z^2}{2}[/imath] instead of [imath]\int_{z^*} = \frac{z^{*2}}{2}[/imath] where [imath]z^*[/imath] is a complex conjugate of [imath]z[/imath]?

 

[blamocur]

Hello,
I have a simple question, why [imath]\int_{z^*} = \frac{z^2}{2}[/imath] instead of [imath]\int_{z^*} = \frac{z^{*2}}{2}[/imath] where [imath]z^*[/imath] is a complex conjugate of [imath]z[/imath]?

The notation does not make sense to me. Which function are you integrating? Do you, by chance, mean an indefinite integral [imath]\int z^{\star} dz[/imath] ? Something else?

 

[Mondo]

@blamocur yes, sorry I made a mistake in formatting this math expression. Of course I meant [imath]\int{z^*}dz[/imath]

PS: Can some mod edit my first post please? Thank you.

 

[blamocur]

I do not believe either is correct. Where did you get this from?

P.S. I do not believe that [imath]\int z^\star dz[/imath] exists. Indeed, if we had [imath]\int z^\star dz = f(z)[/imath] then [imath]f^\prime = z^\star[/imath]. But any differentiable (in complex sense) [imath]f(z)[/imath] is differentiable many times, in particular [imath]f^{\prime\prime}[/imath] must exist, which means the existence of [imath]\frac{d z^\star}{dz}[/imath]. Can you show that [imath]z^\star[/imath] is not differentiable in complex sense?

 

[Mondo]

I was just curious what an integral of complex conjugate is and wolfram gives this https://www.wolframalpha.com/input?i=integral+conjugate(z)+dz
I just noticed this comment "Indefinite integral assuming all variables are real" so the result they give is only when z is real in which case conjugate of z is just z. But why wouldn't it exist? [imath]z = x + iy[/imath] while [imath]z^* = x - iy[/imath] so they are just reflected in the real axis. Why would that cause the integral to cease to exist?

 

[Dr.Peterson]

But why wouldn't it exist? [imath]z = x + iy[/imath] while [imath]z^* = x - iy[/imath] so they are just reflected in the real axis. Why would that cause the integral to cease to exist?

How much do you know about derivatives or integrals of complex functions? I suspect you haven't studied complex variables at all, and just made up the question without knowing what it means.

 

[blamocur]

I was just curious what an integral of complex conjugate is and wolfram gives this https://www.wolframalpha.com/input?i=integral+conjugate(z)+dz
I just noticed this comment "Indefinite integral assuming all variables are real" so the result they give is only when z is real in which case conjugate of z is just z. But why wouldn't it exist? [imath]z = x + iy[/imath] while [imath]z^* = x - iy[/imath] so they are just reflected in the real axis. Why would that cause the integral to cease to exist?

I cannot speak for Wolfram. Have you looked at my post #4 ? I.e., can you differentiate [imath]z^\star[/imath] ?