Analytic Functions
Holomorphic Functions
Complex Differentiability
The limit must exist and be the same regardless of the direction from which (real, imaginary, or any complex direction).
Examples
Example 1: is entire with .
Example 2: is entire with .
Example 3: is nowhere holomorphic (limit depends on direction).
Example 4: is holomorphic only at .
Entire Functions
Properties of Holomorphic Functions
Cauchy-Riemann Equations
Harmonic Functions
Consequences of Cauchy-Riemann
If is holomorphic, then both and are harmonic: and
Analytic Functions
Maximum Modulus Principle
Conformal Mapping
Holomorphic functions with non-zero derivative are conformal (angle-preserving) maps.
Applications
- Fluid dynamics (potential flow)
- Electrostatics
- Heat conduction
- Analytic number theory
- Signal processing (Fourier analysis)