Tag Archives: complex numbers

Swiggety swog, what’s in the blog?

Today we learned how to use complex analysis to solve real-values integrals that would otherwise be very difficult to solve.

Example:

adadada

No complex variables in sight in that integral, right (assuming x is real-valued, haha)? Well you can CONVERT THIS TO A COMPLEX-VALUED INTEGRAL AND HAVE AN EASIER TO SOLVE PROBLEM!

That freaking blew my mind this morning in class. I’d go through the details of how to do this, but I’m a lazyass and don’t want to use Word’s equation editor to make like 30 different equations showing the steps to solve. Instead, I’ll link to Dr. Datta’s notes from class. Go to page 10 in the PDF (the page labeled “161”) for this example.

FREAKING. AWESOME.

Side note: if any of you ever end up going back to UI or know anyone who will be taking some upper-division math classes there, I highly recommend Dr. Datta. She’s very clear at explaining things, good at giving examples, gives reasonable homework, and is always willing to help.

 

De Moivre!

Today in Complex Variables we learned about the de Moivre formula. So, like I do every time I learn a formula named after some dude, I had to look up the dude to see who he was.

Abraham de Moivre lived from 1667 – 1754 (another one of those long-lived mathematicians) and was friends with Newton, Halley, and Stirling, among others.
Originally from France, he moved to London and, while there, became pretty obsessed with Newton’s newly-published Principia Mathematica and basically memorized the material.
In fact, he took Newton’s binomial theorem and was able to generalize it to the multinomial theorem. This work (plus the fact that he was friends with Newton, I’m sure) got him membership in the Royal Society in 1697.

[And—I have to mention it, I’m sorry—he was one of the dudes on Newton’s little crony committee that was put together to hear the plagiarism charges against Leibniz in 1712.]

De Moivre’s famous formula originated (apparently) from this derivation in 1707:

deMoivre

which he later generalized to this form:

deMoivre2

Euler proved it using his own Euler’s formula, so that pretty much cinched it. The reason de Moivre’s formula is so important is because it creates quite a nice connection between complex numbers and trigonometry.

Nifty, eh?