# Complex Integral

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## Main Question or Discussion Point

I feel ashamed asking this, but how do you take the integral of a complex or pure imaginary function?

My sheer guess is that you take the real parts of the function and integrate them seperately, then take the imaginary part and integrate it, but I don't quite know how to do that last part.

Also, can the definite integral of a complex or imaginary function ever be a real number? Does any function have a derivative or integral of i? And just to finish this up, is i positive or negative?

Generally,we substiute the z=x+iy into f(z) and transform it to f(x+iy)=u(x,y)+iv(x,y).So is the dz=dx+idy.Then,we must choose a path connecting A point with B point on the complex plane.Finally,calculate f(z) along the path as you have done in line integrals.However,the result for the same f(z) depends on the path you have choose,that is to say, we can get different results.
If the f(z) is analytical function satisfying Cauchy-Riemann condition, and the path is close ( we may call it contour ,this is the usual condition we face!),the result is generally unique.To work out it,we use the Cauchy formula or theorem of residues.

Also, can the definite integral of a complex or imaginary function ever be a real number?
Yes. The integral of an analytic function on a (suitably nice) closed curve is zero.

is i positive or negative?
Positive and negative are not defined for complex numbers.

I will not that Dyson does a nice job outlining the basic idea of a contour integral. More generally, we may integrate any complex-valued function on a measure space by breaking it into its real and imaginary parts exactly as you describe.

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Thank you for your help, I can see that my intuition was correct.

If I may ask, is there a seperate positive and negative for imaginary numbers, such as 3i being positive while -27i is negative?

No, C is not an ordered field, so you can't define positive numbers or negative numbers.
On the imaginary axis you can define order, since it resembles the real axis, but rarely one speak only of the imaginary axis.

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So you can define order for pure imaginaries?

Yes, it simply the order between the real numbers that multiplies the imaginary unit.

But as I said, I can't find any use in working just in the imaginary axis. (Note that the group of pure imaginaries isn't even closed under multiplication)