Integrating Complex Variables - Types & Solutions

[BeauGeste]

I'm a little confused about integration with complex variables.
Are there two types of integrals?:
1. Integrands with complex numbers but the variable of integration is real.
2. Intregands with complex numbers and the variable of integration is also complex.
But can't (2.) be made into (1.) by dz = dx + i dy...you then have two integrations over real numbers...?

 

[SiddharthM]

real numbers are complex numbers.

integration is often done on a contour in the plane, one could parametrize this contour and use real integration like you suggested. It would prolly be tougher to do this tho.

 

[BeauGeste]

how would you integrate a complex function over all space (whole complex plane)? integrate real axis -inf to +inf and then integrate imaginary axis -inf to +inf? Or do a contour integral where the contour is a circle and extend the radius to infinity?

 

[Vampyre747]

I think that the first way would be more difficult if possible. You would just use a contour integral over a circle with a radius going to infinity.

 

[robert Ihnot]

SiddharthM: Real numbers are complex numbers.

This echoes my friend's engineering Prof. He explained to his Freshman class that Mathematicians made an unfortunate choice of words, and "Complex numbers are as real as real numbers."

 

[FunkyDwarf]

The reals are a subset of the complex numbers is probably a better way to word it.

 

[HallsofIvy]

how would you integrate a complex function over all space (whole complex plane)? integrate real axis -inf to +inf and then integrate imaginary axis -inf to +inf? Or do a contour integral where the contour is a circle and extend the radius to infinity?

When you are integrating real valued functions of real numbers you typically integrate from one number to another. Since complex numbers correspond to points in the PLANE, to do the equivalent you need to specify a path between the two points. If you really mean integrate over the entire complex plane, they you let z= x+ iy and do a double integral over x and y.