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Lecture

Laplace Transform: Definition and Examples

In course

MATH-207(a): Analysis IV (for SV, MT)

Ce cours est une introduction à la théorie des séries de Fourier, transformées de Fourier (y compris pour les distributions tempérées), à la transformée de Laplace, et à leurs utilisations pour résoudre des équations différentielles ordinaires et aux dériv ...

Description

This lecture introduces the Laplace transform as a tool to analyze functions, connecting it to the Fourier transform. The instructor explains the formal definition of the Laplace transform, its convergence properties, and provides examples of calculating the Laplace transform for constant and exponential functions. The lecture emphasizes the importance of the convergence abscissa in determining the validity of the Laplace transform calculation, showcasing how it differs from the Fourier transform. Through detailed mathematical derivations, the instructor illustrates the step-by-step process of finding the Laplace transform for different types of functions, highlighting the conditions for convergence and the resulting transformed functions.

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Related concepts (23)

Fourier transform

In physics and mathematics, the Fourier transform (FT) is a transform that converts a function into a form that describes the frequencies present in the original function. The output of the transform is a complex-valued function of frequency. The term Fourier transform refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made the Fourier transform is sometimes called the frequency domain representation of the original function.

Integral transform

In mathematics, an integral transform maps a function from its original function space into another function space via integration, where some of the properties of the original function might be more easily characterized and manipulated than in the original function space. The transformed function can generally be mapped back to the original function space using the inverse transform. An integral transform is any transform of the following form: The input of this transform is a function , and the output is another function .

Laplace transform

In mathematics, the 'Laplace transform, named after its discoverer Pierre-Simon Laplace (ləˈplɑ:s), is an integral transform that converts a function of a real variable (usually , in the time domain) to a function of a complex variable (in the complex frequency domain, also known as s-domain', or s-plane). The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms ordinary differential equations into algebraic equations and convolution into multiplication.

Discrete Fourier transform

In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. The interval at which the DTFT is sampled is the reciprocal of the duration of the input sequence. An inverse DFT (IDFT) is a Fourier series, using the DTFT samples as coefficients of complex sinusoids at the corresponding DTFT frequencies.

Fourier analysis

In mathematics, Fourier analysis (ˈfʊrieɪ,_-iər) is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Joseph Fourier, who showed that representing a function as a sum of trigonometric functions greatly simplifies the study of heat transfer. The subject of Fourier analysis encompasses a vast spectrum of mathematics.

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