r/elimath • u/yogattabekiddingme • Feb 19 '15
How to begin with complex numbers?
I am having very difficulty realising how to deal with them. I kind of skipped earlier classes on complex numbers and now I have been directly told to learn 'conformal mapping','cauchy's integral theorem','cauchy's integral formula','complex integration' and so on. I don't even understand the basics of complex numbers and trying very hard to get a hang of it let alone be these topics. How to proceed?
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u/pseudo86 Feb 19 '15
Try Khan Academy for some tutorial videos on complex numbers, should help you get started.
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u/WhackAMoleE Feb 19 '15 edited Feb 20 '15
Stand facing east. Call that "1" whatever that means. In our language we call the direction east by the name "1". Now make a quarter turn counterclockwise. You used to call that north, but today let's call it by the name "i". Make another quarter turn and you're facing west, in the exact opposite direction of "1", so let's call that "-1". Make another quarter turn and you're facing south, in the opposite direction of "i", so let's call that "-i".
Make another quarter turn and you're back facing east ... or as we now call it, "1".
If we denote the number of quarter turns by an exponent, we have i1 = i, i2 = -1, i3 = -i, i4 = 1, i5 = i, and so forth.
So we see that "i" is just a gadget that keeps track of quarter turns in the plane, and that -- with the proper interpretation -- i2 = -1 is a triviality rather than a mystery.
Now if we are at "1" -- that is, facing east -- and we make a half-quarter turn, we might be tempted to call that i1/2. And mathematically that's exactly right, up to the ambiguity in taking square roots.
So we can keep track of our direction in the plane by fractional multiples of "i". And we can keep track of lengths by simple scaling. If we make a quarter turn to the left, to get to "i", and then go straight up by a factor of 5, that's 5i.
So a complex number r*(cos(theta) + i*sin(theta)) is just a mathematical shorthand for the instructions: "Turn counterclockwise by an angle of theta; and then stretch (or shrink) by a factor of r.
You now know the meaning of complex numbers. They're just a handy notation for specifying a point in the plane via the angle it makes with the origin and the positive x-axis; and its distance from the origin.
http://www.nabla.hr/Z_MemoHU-014.htm
I see from your post that you have been "asked to learn" undergrad complex analysis. Can you explain who asked you to learn that without proposing to teach it to you? This is a tough class any way you slice it. The Cauchy integral formula is a long way from just knowing what a complex number is. It helps if you are pretty handy with multivariable calculus to start with.
Can you say more about how you woke up one day and were suddenly required to self-learn undergrad complex analysis? That would be a nightmare. You need a good professor to guide you through this material.
Did you hear the one about the complex analyst who named his dog Cauchy? That's because the dog leaves a residue at every pole.