so suppose i wanted to calculate the antiderivatives of [itex]e^x\sin{x}[/itex] and just for the hell of it also [itex]e^x\cos{x}[/itex]. 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.(adsbygoogle = window.adsbygoogle || []).push({});

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

[tex]e^{(1+i)x}=e^{x+ix}=e^x\cos{x}+ie^x\sin{x}[/tex]

and leave off the limits

[tex]\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)[/tex]

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

[tex]\int e^x \cos{x} = \frac{1}{2}e^x(\cos{x}+\sin{x})[/tex]

[tex]\int e^x \sin{x} = \frac{1}{2}e^x(\sin{x}-\cos{x})[/tex]

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?

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# Perverting a contour integral

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