Laplace's Method: Exercises

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Complex Exponential: De Moivre's Formula

Covers De Moivre's formula for finding roots of complex numbers and the concept of complex exponential.

Harmonic Forms and Riemann Surfaces

Explores harmonic forms on Riemann surfaces, covering uniqueness of solutions and the Riemann bilinear identity.

Solving the Quintic: Dominant Balance Analysis

Explores dominant balance analysis in solving the quintic polynomial, revealing insights into root behavior and the importance of symbolic expressions.

Complex Analysis: Taylor Series

Explores Taylor series in complex analysis, emphasizing the behavior around singular points.

Complex Analysis: Laurent Series

Explores Laurent series in complex analysis, emphasizing singularities, residues, and the Cauchy theorem.

Holomorphic Functions: Cauchy-Riemann Equations and Applications

Discusses holomorphic functions, focusing on the Cauchy-Riemann equations and their applications in complex analysis.

Complex Integration: Fourier Transform Techniques

Discusses complex integration techniques for calculating Fourier transforms and introduces the Laplace transform's applications.

Power Series Computations

Covers power series, generating functions, and operations like addition, multiplication, differentiation, and integration, with examples and the generalized binomial theorem.

Taylor Polynomials: Calculating Limits and Derivatives

Covers the calculation of Taylor polynomials and their applications in limits and derivatives of functions from R² to R.

Poisson Problem: Fourier Transform Approach

Explores solving the Poisson problem using Fourier transform, discussing source terms, boundary conditions, and solution uniqueness.