Integrating a function of the complex conjugate of x with respect to dx

HilbertSpace

The reason I ask the aforementioned question is because I came across the expectation values of operators in Quantum Mechanics. And part of the computation involves integrating a function of the complex conjugate of x with respect to dx.

So as an example let's say I have:

∫ sin (x*) dx where the * denotes the complex conjugate.

How would I do it? (I would post an attempt to solve it but I have no idea whether to even treat the x* as a constant, or variable or.....something).


Thank you in advance for any help you may provide.

Muphrid

Is [itex]x[/itex] a complex variable, then?

HilbertSpace

Yes, x is a complex variable.

Skrew

If your x is just real or imaginary conjugate(x) = x or conjugate(x) = -x and so integration can be performed pretty easily.

I don't believe the function sin(conjugate(z)) is holomorphic so it won't end up having a complex anti-derivative.

This mean that numerical integration will be needed.

algebrat

Is it a definite or indefinite integral?

If it is definite, what path would you like to use? A complex integral can be looked at as a path integral in the plane, for which a parametrization might make it feasible to calculate exactly.

But like Skrew said, this does not look amenable to methods from complex analysis like computing residues.

Mandlebra

IN Fact holomorphic depends on the derivative wrt z* to be zero

Skrew

I'm not well versed on different approaches but my book defines d/dz and d/d(conjugate(z)) to be such that the cauchy Riemman equations are satisified IFF d/d(conjugate(z)) = 0, at which point the definition of the derivative can be applied and the limit exists.

Is there another way to look at this using the difference quotient of conjugate(z)?

haruspex

Some insight may be gained by considering the integral over a path P consisting of a straight line segment at angle θ to the real axis:
[itex]\int_{P}f(z).d\overline{z} = e^{2i\theta} \int_{P}f(z).dz[/itex]