Perverting a contour integral

1. Mar 5, 2009

ice109

so suppose i wanted to calculate the antiderivatives of $e^x\sin{x}$ and just for the hell of it also $e^x\cos{x}$. well i could perform integration by parts twice recognize that the original integral when it reappears, subtract from one side to the other blah blah blah.

or i could pervert a contour integral: i could integrate $e^z$ along the line in the $(1,i)$ direction:

$$e^{(1+i)x}=e^{x+ix}=e^x\cos{x}+ie^x\sin{x}$$

and leave off the limits

$$\int e^{(1+i)x}dx=\frac{1}{1+i}e^{(1+i)x}=\frac{1}{2}\left(e^x(\cos{x}+\sin{x})+ie^x(\sin{x}-\cos{x})\right)$$

and compare the real and imaginary parts of the integrand and the "antiderivative" and conclude:

$$\int e^x \cos{x} = \frac{1}{2}e^x(\cos{x}+\sin{x})$$
$$\int e^x \sin{x} = \frac{1}{2}e^x(\sin{x}-\cos{x})$$

which are of course the correct answers.

what i don't know is what an indefinite integral in the complex plane even is. i've only so far learned that i can use this "trick" to perform contour integrals <=> with end points, and then make conclusions about definite integrals of the real and imaginary parts of the integrand.

so how legit is this?

2. Mar 5, 2009

lurflurf

That is convoluted.
let s be the antiderivative opperator
s*exp(x)cos(x)=Re[s*exp(x(1+i))]
s*exp(x)cos(x)=Im[s*exp(x(1+i))]
s*exp(x(1+i))=exp(x(1+i))/(1+i)

3. Mar 5, 2009

ice109

and how do i know s commutes with Re and Im?

4. Mar 5, 2009

HallsofIvy

Staff Emeritus
Surly "perverted" is not the right word here!

5. Mar 5, 2009

ice109

can one of you more knowledgeable people just tell me if this a legitimate technique?

6. Mar 5, 2009

Ben Niehoff

It is legitimate for entire functions. There may be subtleties if a function has poles.

7. Mar 5, 2009

dvs

A function f:R->C can be decomposed into its real and imaginary parts, say f=u+iv, where u=Re(f) and v=Im(f). The definition of the Riemann/Lebesgue integral of such a function is

$$\int f = \int u + i \int v.$$

The integrals $\int u$ and $\int v$ are just the usual real integrals.

Does this clear anything up for you?