Cauchy formula for repeated integration
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The Cauchy formula for repeated integration allows one to compress n antidifferentiations of a function into a single integral.
[edit] 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).
[edit] References
- Gerald B. Folland, Advanced Calculus, p. 193, Prentice Hall (2002). ISBN 0-13-065265-2