What are the Applications of Complex Analysis in Calculus?

[grief]

for our project in calculus, I am doing a presentation on the basics of complex analysis. Somewhere along there I need to tackle the question: what are the applications of complex analysis?
Are there any application problems that I can give that involve basic derivatives/integrals of complex functions?

 

[coalquay404]

for our project in calculus, I am doing a presentation on the basics of complex analysis. Somewhere along there I need to tackle the question: what are the applications of complex analysis?
Are there any application problems that I can give that involve basic derivatives/integrals of complex functions?

Complex analysis appears *everywhere*. It might be beyond the level of what you want, but a pretty neat example might be to look at the Feynman propagator for the Klein-Gordon field. That should be a pretty good way to demonstrate contour integrals and the residue theorem, as well as giving a physically-relevant application.

It might also be worth looking at some spacetimes. Think about making a Euclidean continuation of Schwarzschild, for example. That'll give you a really cool way to derive the temperature of the Schwarzschild black hole.

 

[grief]

help! I don't understand! besides this has to be a 30 mimute presentation for a bunch of tired high school seniors (ie our class) who haven't seen the square root of negitive one for two years.

 

[coalquay404]

Ah, well in that case an obvious example is to look at something like wave motion. Try to google around for an application of complex analysis to oscillating systems and you'll find plenty of examples of complex numbers being used.

 

[grief]

thank you, It's still hard to find something online, but I guess I'll just mention it quickly in the presentation

 

[mathwonk]

both real and the imaginary parts of a complex differentiable functions satisfy laplaces equation, the equation for a steady state heat distribution.and if you want mathematical applications, the euler product formula leads to the riemann zeta function that gibves a nice proof of the infinitude of primes.

 

[reilly]

Standard RLC circuits would work for you.
Regards,
Reilly Atkinson