Mathematical_Analysis

 Also a student of Lagrange and Laplace:

                                                                  

Formule sommatoire de Poisson — Wikipédia (wikipedia.org)

First showed how one could work more easily from a transform than from its

original function.

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The modern notion of the fourier transform was articulated by Henri Lebesgus (1875-1941).

From Copilot:

The Fourier Transform is primarily attributed to Jean-Baptiste Joseph Fourier, who introduced the concept of representing functions as sums of sines and cosines, known as Fourier series. However, Henri Lebesgue made significant contributions that extended and refined Fourier’s work.

Lebesgue is renowned for developing Lebesgue integration, which provided a more robust framework for integrating functions, especially those that are not necessarily continuous1. This was crucial for the advancement of Fourier analysis, as it allowed for a more comprehensive understanding and manipulation of Fourier series and transforms2.

Lebesgue’s work also led to the formulation of the Riemann-Lebesgue Lemma, which states that the Fourier coefficients of an integrable function tend to zero3. This lemma is fundamental in harmonic analysis and the study of Fourier series.

In summary, while we owe the original concept of the Fourier Transform to Joseph Fourier, Henri Lebesgue’s contributions were pivotal in generalizing and formalizing the mathematical foundations that underpin modern Fourier analysis.

                                              

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Had to ask: How do modern computers calculate integrals?

Copilot:

Modern computers calculate integrals using a combination of numerical methods and symbolic computation.

Numerical Methods

Numerical integration involves approximating the value of an integral using discrete data points. Some common numerical methods include:

1. Trapezoidal Rule: Approximates the area under the curve as a series of trapezoids.
2. Simpson’s Rule: Uses parabolic segments to approximate the area under the curve.
3. Gaussian Quadrature: Selects specific points and weights to achieve high accuracy with fewer evaluations1.

Symbolic Computation

For symbolic integration, computers use algorithms to manipulate mathematical expressions symbolically. Some well-known algorithms include:

• Risch Algorithm: Used for integrating elementary functions.
• Gröbner Bases: Helps in solving systems of polynomial equations2.

Hybrid Approaches

Many modern systems, like Mathematica and MATLAB, combine both numerical and symbolic methods to provide accurate and efficient solutions.

MIT School of Engineering | » How do computers perform complex mathematical operations?

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Euler's formula:   e^ix = cos(x) + i*sin(x), was published in 1748. He used the power

series from sine and cosine for hos proof.

These series were known to Isaac Newton (1642-1727).

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Brook Taylor (1685 - 1731) - Biography - MacTutor History of Mathematics (st-andrews.ac.uk)