Encyclopedia > Cauchy formula for repeated integration
The Cauchy formula for repeated integration allows one to compress nantidifferentiations of a function into a single integral. Augustin Louis Cauchy Augustin Louis Cauchy (August 21, 1789 – May 23, 1857) was a French mathematician. ... In calculus, an antiderivative or primitive function of a given real valued function f is a function F whose derivative is equal to f, i. ...
Scalar case
Let f be a continuous function on the real line. Then the nth antidifferentiation of f,
,
is given by single integration
.
A proof is given by induction. Since f is continuous, the base case is given by
.
A little work shows that we also have
.
Hence, f[n](x) gives the nth antidifferentiation of f(x).
References
Gerald B. Folland, Advanced Calculus, p. 193, Prentice Hall (2002). ISBN 0-13-065265-2
In mathematics, Leibniz formula for π, due to Gottfried Leibniz, states that Proof Consider the infinite geometric series It is the limit of the truncated geometric series Splitting the integrand as and integrating both sides from 0 to 1, we have Integrating the first integral (over the truncated geometric series) termwise...
For fast calculations, one may use formulae such as Machin's: John Machin, (1680—June 9, 1751), a professor of astronomy in London, is best known for developing a quickly converging series for π in 1706 and using it to compute π to 100 decimal places.
This formula is most easily verified using polar coordinates of complex numbers, starting with As the degree of the Taylor series rises, it approaches the correct function.
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