This course served as an introduction to the theory of complex analysis, Fourier series and Fourier transforms, the Laplace transform, with applications to the theory of ordinary and partial differential equations. These tools play an integral role in most branches of science and engineering.

Contents

Complex analysis:

• Definitions and examples of complex functions.
• Holomorphic functions.
• Cauchy-Riemann equations.
• Complex integrals and Cauchy formulas.
• Series of Laurent.
• Residue theorem.

Laplace’s analysis:

• Laplace transforms.
• Fourier transforms.
• Applications to ordinary differential equations.
• Applications to partial differential equations.