How to find conformal mappings between different domains in complex analysis?

**Map from unit disk to upper half-plane:** $$w = i \cdot \frac{1 - z}{1 + z}$$ This is a Möbius transformation. It sends $-1 \to \infty$, $1 \to 0$, $-i \to 1$, mapping the disk $|z| < 1$ to $\text{Im}(w) > 0$. **General Möbius Transformation:** $$w = \frac{az + b}{cz + d}, \quad ad - bc \ eq 0$$ Properties: - Maps circles/lines to circles/lines - Preserves angles (conformal everywhere except $z = -d/c$) - Determined by 3 points: choose 3 points in the domain and map them to 3 points in the target **Map from strip $0 < \text{Im}(z) < \pi$ to upper half-plane:** $$w = e^z$$ Since $z = x + iy$, $e^z = e^x e^{iy}$. As $y$ goes from 0 to $\pi$, $e^{iy}$ sweeps the upper half of the unit circle. Combined with $e^x$ (which gives all positive radii), this maps the strip onto $\text{Im}(w) > 0$. **Riemann Mapping Theorem:** Any simply connected proper subset of $\mathbb{C}$ can be conformally mapped onto the unit disk. This is a **pure existence theorem** — it doesn't construct the map, but it guarantees one exists. This is powerful because it means we can solve problems (like fluid flow or electrostatics) on complicated domains by mapping them to simpler ones. **Common conformal maps to remember:** - $w = z^2$: maps quarter-plane to half-plane - $w = \sqrt{z}$: maps half-plane to quarter-plane - $w = \log z$: maps angular sector to strip - $w = \frac{z-1}{z+1}$: maps right half-plane to unit disk