Integral Transforms - 3PMCTIN9

Goal(s)

Presentation of basic mathematical concepts required for computing and using Fourier Transform and Laplace Transform. The first section is devoted to the presentation of the analytic functions of a complex variable and to the integration of a complex-valued function along a curve in the complex plan. The second section presents the expansion of a periodic function in Fourier Series and the Fourier Transform of absolutely summable functions. In the third section the Laplace Transform is explained. The usability of these Transforms in solving differential equations is also presented.

Content(s)

I. Analytic functions of the complex variable
Derivability; Cauchy-Riemann Conditions; Holomorphic functions; Singular Points; Integral along a curve in the complex plan; Jordan’s Lemma; Cauchy Theorem; Residue theorem.
II. Fourier Transform
Fourier Series expansion of periodic functions; Dirichlet Condition; Fourier Transform of absolutely summable functions; Inverse Fourier Transform; Properties of the Fourier Transform; Convolution.
III. Laplace Transform of causal functions
Causal functions ; Laplace Transform of causal functions ; Properties of Laplace Transform ; Inverse Laplace Transform ; Solving differential equations.

Complex variables; Integration; Taylor series – Level Bachelor 2nd year