"Odd" Integrals

[Drake Glau]

Not in the curriculum at all I think, just a random thought I had during my calculus final and I figured if anyone knew it'd be the HL kids perhaps...

The integral of i

anyone have any idea on this? I tried googling it and get someone else and was wondering if anyone knew anything on it. Just kind of interested

[aldld]

Well when you think of it, i is just a constant and could probably be treated as such. In fact, if you want to look at calculus involving complex functions, you may be interested in looking at complex analysis.

[Drake Glau]

So if I had |idx (used | for the integral sign) I would just end up with ix?

I've never really studied complex numbers at all so I wasn't sure how you could treat it but now that I'm starting to fully understand pretty much every function in the real world I found it interesting at how those functions would act in a different world, that make sense?

[aldld]

So if I had |idx (used | for the integral sign) I would just end up with ix?

I've never really studied complex numbers at all so I wasn't sure how you could treat it but now that I'm starting to fully understand pretty much every function in the real world I found it interesting at how those functions would act in a different world, that make sense?

Sort of. It's actually not completely obvious how to do calculus with complex numbers, especially since we're used to real numbers.

Disclaimer: I have not studied any complex analysis at all and only a very small amount of multivariable calculus. But here's how I would think of this: In single variable calculus, you integrate some function over some interval of the real line. In multivariable calculus you can generalize this idea to integrating perhaps over an arbitrary curve in the plane or some other region of the plane. I imagine there would be a similar deal with some aspects of complex functions, where you integrate over some region of the complex plane.

Of course, I'm almost certain there are many more subtleties to it than just that. But yeah.

[CocoPop]

If you do maths/engineering/physics/chemistry at uni there will be a lot of calculus involving imaginary or complex numbers. Remember that i does not vary (di = 0), so you treat it as a constant in calculus.

EDIT: In my experience, when it comes to visualising complex functions, one tends to draw two separate plots - one for magnitude and the other for phase.

So for example, if you had the function f(x) = x/(i + x), you'd have:

arg(f) = arctan(0) - arctan(1/x) = - arctan(1/x)

You can plot that against x and you'll find that the phase varies from -90 degrees (at x = 0) to 0 degrees (as x-> infinity).

|f(x)| = x/sqrt(1 + x^2)

It's very common to use this method when looking at frequency responses or transfer functions of circuits. For example, if x is the frequency of an AC system, it lets you see the amplitude and phase of the different frequency components in a signal.

Edited June 3, 2012 by CocoPop