🔢 Complex Functions and Special Functions

Learn basic operations of complex numbers, polar form representation on complex plane, Euler's formula, and implement visualization techniques for complex functions. Understand geometric meaning of complex numbers and introduce applications to complex impedance and crystal structure analysis in materials science.

Learn differentiability of complex functions, Cauchy-Riemann relations, properties of analytic functions. Implement calculation methods of complex integrals and Cauchy's integral theorem, and cover applications to potential theory and fluid dynamics.

Learn Taylor series expansion and Laurent series expansion, classification of singularities (removable singularity, pole, essential singularity). Implement residue calculation methods and residue theorem, and cover evaluation of real integrals and applications to physics problems.

Learn theory and numerical implementation of Fourier series, Fourier transform, and Laplace transform. Cover frequency analysis, convolution theorem, filtering, and solving differential equations, and implement applications to signal processing and spectral analysis.

Learn special functions such as Gamma function, Bessel function, Legendre polynomials, and Hermite polynomials. Implement solving differential equations in cylindrical and spherical coordinate systems, applications to boundary value problems, properties and numerical computation of orthogonal polynomials.