Title: Part IB / Complex Analysis
Tags: Complex_Differentiation

Q: Definition. A path in \(U \subseteq \mathbb{C}\) is [...]; we say \(C^1\) if \(\gamma'\) exists and is continuous.
A: a continuous map \(\gamma : [a,b] \to U\)

Q: Definition. A path is simple if [...]
A: it is injective

Q: Definition. \(U \in \mathbb{C}\) is path-connected if [...]
A: \(\forall z,w \in U\), there exists a path in U connecting z to w.
Back Extra: If U is open, z,w, connected by a path \(\gamma\) in U, then there exists a path P in U consisting of finitely many horizontal and vertical segments. This follows from the compactness of an interval [a,b].

Q: Definition. A domain in \(\mathbb{C}\) is [...]
A: an open, non-empty, path-connected subset.

Q: Definition. \(f : U \to \mathbb{C}\) on a domain U is differentiable at \(u \in U\) if [...]
A: \[f'(u) = \lim_{z \to u} \frac{f(z) - f(u)}{z-u} \text{ exists}\]

Q: Definition. \(f : U \to \mathbb{C}\) is holomorphic at \(u \in U\) if [...]
A: \(\exists \epsilon > 0\) s.t. \(\forall z \in \mathbb{D}(u , \epsilon)\), f is differentiable at z.
Back Extra: This is also called analytic.

Q: Definition. \(f : \mathbb{C} \to \mathbb{C}\) is entire if [...]
A: it is holomorphic on \(\mathbb{C}\).

Q: Proposition. Cauchy-Riemann equations. Let \(f: U \to \mathbb{C}\) on an open set \(U \subseteq \mathbb{C}\). Then f is differentiable at \(w = c + id\) in U with \(f'(w) = p + iq\) iff, writing \(f = u + iv\), we have u,v are real differentiable at \((c,d)\) and \[u_x = v_y \; \; \; -u_y = v_x\]
A: f differentiable at \(w = c + id\) with \(f'(w) = p + iq\)  \[\iff \lim_{z \to w} \frac{f(z) - f(w) - (z-w)(p + iq)}{z -w} = 0\] \[\iff \lim_{z \to w} \frac{f(z) - f(w) - (z-w)(p + iq)}{|z-w|} = 0\] Writing \(f = u + iv\) and considering real and imaginary parts, this holds iff both of \[\lim_{(x,y) \to (c,d)} \frac{u(x,y) - u(c,d) - [p(x-c) - q(y-d)]}{\sqrt{(x-c)^2 + (y-d)^2}} = 0\] and \[\lim_{(x,y) \to (c,d)} \frac{v(x,y) - v(c,d) - [q(x-c) + p(y-d)]}{\sqrt{(x-c)^2 + (y-d)^2}} = 0\] from this we can see that \(u_x = p = v_y, u_y = -q = -v_x\)

Q: Definition. U,V domains in \(\mathbb{C}\), \(f : U \to V\) a map is a conformed equivalence if [...]
A: f is a bijective holomorphic map with \(f'(z) \neq 0 \; \forall z \in U\)
Back Extra: We say U and V and conformally equivalent if such a map exists.