PHYS20672 Summary 1

Complex numbers

1. Complex numbers in the form $x+\ii y$ and $r \e^{\ii\theta}$; complex plane and representation by 2D vectors
2. Fundamental Theorem of Algebra: algebraic equation of degree $n$ as $(z-z_1)(z-z_2)\ldots(z-z_n)=0$
3. The $n$th roots of unity: $\e^{2\pi\ii k/n}$ for $0\le k < n$
4. Sets of points; interior, boundary and exterior points; open sets; curves and regions of the complex plane.

Spiegel 1.1-1.5, 1.7, 1.8, 1.10-1.14, 1.18

Riley 2; Boas 2; Arfken 6.1

Functions of complex variables

1. Real and imaginary parts of functions: $f(z)=u(x,y)+\ii v(x,y)$
2. Standard functions - ratios of polynomials, exponential and log, trig and hyperbolic trig functions
3. Multiple-valued functions; non-integer powers and log; principal values of functions, e.g. $\Arg z$ versus $\arg z$ and $\Ln z$ versus $\ln z$; branches and branch cuts

Spiegel 2.1-2.7

Riley 18.1, 18.5; Boas 14.1; Arfken 6.1

Functions as mappings

1. Mappings of points, curves and regions from the $z$ plane to the plane of $w=f(z)$
2. Curves which circle $z_0$ in the $z$-plane map to curves which circle the origin in the $w$ plane if $f(z_0)=0$
3. The argument theorem: $\Delta\phi=2\pi N$, where $\phi=\arg w$ and $N$ is the number of simple zeros of $f$ that are enclosed by a curve in the $z$-plane

In the diagrams above the blue and red lines are the mappings of the lines $x=$ const and $y=$ const, and the black dot is the mapping of the point $z=0$.

Spiegel 2.4-2.7

(Riley 14.8); (Boas 14.9); Arfken 6.6

Differentiation and Cauchy-Riemann equations

1. Definition of derivative: $$\dby fz=\lim_{\delta z\to 0}\frac{f(z+\delta z)-f(z)}{\delta z}$$
2. Derivative must be finite and independent of direction
3. Analytic (regular, holomorphic) functions
4. Cauchy-Riemann equations $$\pdby ux=\pdby vy\quad\text{and}\quad \pdby uy=-\pdby vx$$
5. Standard functions analytic over their domain of definition
6. Usual rules apply, e.g. chain and product rule, $\dslby{(\sin z)}z=\cos z$, etc.
7. Existence of all higher derivatives
8. Functions $u(x,y)$ and $v(x,y)$ are conjugate; one can be reconstructed from the other
9. Functions $u(x,y)$ and $v(x,y)$ are harmonic (obey Laplace's equation)

Riley 18.1, 18.2; Boas 14.2; Arfken 6.2