My textbook doesn't cover this, that I'm aware, and I must have missed it in lecture. It's a practice exam question, which means it's meant to be done fairly quickly. But I don't see any easy or quick ways to do this.Originally Posted by practice exam
Any help would be much appreciated!
and . So I guess I could do
. It follows that has a Taylor series about any , and that .
Is that the easiest way to do this though? I suspect that I knew to jump from to because I was expecting it based on Drexel28's observation. But on an exam that might not be quite so obvious.
Thanks for the help guys!
Also, it might help to remember why we have the "Cauchy-Riemann" equations.
Suppose f(z)= u(x,y)+ iv(x,y) is to be differentiated at . Of course,
but because the complex plane is two dimensional, for that limit to exist, we must get tthe same limit taking any path toward . In particular, if we approach along a line parallel to the real axis, we have
While if we approach along a path parallel to the imaginary axis
In order that f be differentiable at , those must be the same so, setting real part equal to real part and imaginary part equal to imaginary part, we have
Those also tell us how to calculate - use either