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Like the integral of the first kind, the complete elliptic integral of the second kind can be computed very efficiently using the arithmetic–geometric mean. [1] Define sequences a n and g n, where a 0 = 1, g 0 = √ 1 − k 2 = k ′ and the recurrence relations a n + 1 = a n + g n / 2 , g n + 1 = √ a n g n hold.
It was clear that those so called elliptic integrals could not be solved using elementary functions. Fagnano observed an algebraic relation between elliptic integrals, what he published in 1750. [13] Euler immediately generalized Fagnano's results and posed his algebraic addition theorem for elliptic integrals. [13]
The duplication theorem can be used for a fast and robust evaluation of the Carlson symmetric form of elliptic integrals and therefore also for the evaluation of Legendre-form of elliptic integrals. Let us calculate R F ( x , y , z ) {\displaystyle R_{F}(x,y,z)} : first, define x 0 = x {\displaystyle x_{0}=x} , y 0 = y {\displaystyle y_{0}=y ...
An even larger, multivolume table is the Integrals and Series by Prudnikov, Brychkov, and Marichev (with volumes 1–3 listing integrals and series of elementary and special functions, volume 4–5 are tables of Laplace transforms).
In 1995, Alan Jeffrey published his Handbook of Mathematical Formulas and Integrals. [22] It was partially based on the fifth English edition of Gradshteyn and Ryzhik's Table of Integrals, Series, and Products and meant as an companion, but written to be more accessible for students and practitioners. [22] It went through four editions up to 2008.
A different technique, which goes back to Laplace (1812), [3] is the following. Let = =. Since the limits on s as y → ±∞ depend on the sign of x, it simplifies the calculation to use the fact that e −x 2 is an even function, and, therefore, the integral over all real numbers is just twice the integral from zero to infinity.
From the 1820s to the 1840s, analytic topics such as elliptical integrals were introduced to the curriculum. Under William Whewell, the Tripos' scope changed to one of 'mixed mathematics', with the inclusion of topics from physics such as electricity, heat and magnetism. Students would have to study intensely to perform routine problems rapidly.
In integral calculus, Euler's formula for complex numbers may be used to evaluate integrals involving trigonometric functions. Using Euler's formula, any trigonometric function may be written in terms of complex exponential functions, namely e i x {\displaystyle e^{ix}} and e − i x {\displaystyle e^{-ix}} and then integrated.