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 $x$ 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]

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.

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

 

[Skrew]

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

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:
$\int_{P}f(z).d\overline{z} = e^{2i\theta} \int_{P}f(z).dz$