Some Complex Stuff
Random number: 66
9th December 2016
I should be studying, but I feel like I should know at least some complex analysis, so here's some very bad preliminary notes, just so I can get a "feel" of it.
So like complex numbers are like \(z = x + iy\) where \(x,y \in \mathbb{R}\) and \(i^2 = -1\). So we have a couple of functions defined for complex numbers:
$$e^z = e^x (\text{cos}(y) + i \text{sin} y)$$
$$\text{cos} (z) = \frac{e^{iz} + e^{-iz}}{2}$$
$$\text{sin} (z) = \frac{e^{iz} - e^{-iz}}{2i}$$
$$\text{log} (z) = \text{ln} |z| + i \text{arg}(z)$$
Where \(\text{arg}\) gives the angle of the complex vector (sort of). Of course, there could be heaps of angles for a single complex number, so there's a 'principle branch of log', which uses \(\text{Arg}(z)\), defined in the range \([-\pi,\pi]\).
Derivatives and Integration is defined as normally, and all the normal rules apply. There is one caveat. Suppose we have the complex function:
$$f(z) = u(x,y) + iv(x,y)$$
for \(f\) to be differentiable at \(z_0\), we need to have the following equations hold (Cauchy-Riemann Equations)
$$\frac{\delta u}{\delta x} (x_0, y_0) = \frac{\delta v}{\delta y} (x_0, y_0)$$
$$\frac{\delta u}{\delta y} (x_0, y_0) = - \frac{\delta v}{\delta x} (x_0, y_0)$$
This comes from taking the partial of \(f\) in the \(x\) direction and the \(y\) direction.
Anyway, skipping forward to some cool stuff. Suppose that \(f\) is analytic in a region containing a closed curve \(C\) with positive orientation, and suppose that \(z_0\) is inside \(C\). Then it turns out that
$$f(z_0) = \frac{1}{2 \pi i} \int_C \frac{f(z)}{z-z_0} dz$$
In fact we have that if \(C_1\) and \(C_2\) are 'homotopic' closed curves (loosely meaning that they can be continuously deformed into one another), then we have that
$$\int_{C_1} f(z) dz = \int_{C_2} f(z) dz$$
and if \(f\) has no holes inside \(C\),
$$\int_{C} f(z) dz = 0$$
Anyway, back to studying.