Smoluchowski coagulation equation

The Cauchy formula for repeated integration, named after Augustin Louis Cauchy, allows one to compress n antidifferentiations of a function into a single integral (cf. Cauchy's formula).

Scalar case

Let ƒ be a continuous function on the real line. Then the nth repeated integral of ƒ based at a,

$f^{(-n)}(x)=\underset{a}{\overset{x}{\int }}\underset{a}{\overset{{\sigma }_1}{\int }}\cdots \underset{a}{\overset{{\sigma }_{n-1}}{\int }}f({\sigma }_n)\mathrm{d}{\sigma }_n\cdots \mathrm{d}{\sigma }_2\mathrm{d}{\sigma }_1$,

is given by single integration

$f^{(-n)}(x)=\frac{1}{(n-1)!}\underset{a}{\overset{x}{\int }}{(x-t)}^{n-1}f(t)\mathrm{d}t$.

A proof is given by induction. Since ƒ is continuous, the base case follows from the Fundamental theorem of calculus:

$\frac{\mathrm{d}}{\mathrm{d}x}{f}^{(-1)}(x)=\frac{\mathrm{d}}{\mathrm{d}x}\underset{a}{\overset{x}{\int }}f(t)\mathrm{d}t=f(x)$;

where

$f^{(-1)}(a)=\underset{a}{\overset{a}{\int }}f(t)\mathrm{d}t=0$.

Now, suppose this is true for n, and let us prove it for n+1. Apply the induction hypothesis and switching the order of integration,

$\begin{array}{rl}{f}^{-(n+1)}(x)& =\underset{a}{\overset{x}{\int }}\underset{a}{\overset{{\sigma }_1}{\int }}\cdots \underset{a}{\overset{{\sigma }_n}{\int }}f({\sigma }_{n+1})\mathrm{d}{\sigma }_{n+1}\cdots \mathrm{d}{\sigma }_2\mathrm{d}{\sigma }_1\\ & =\frac{1}{(n-1)!}\underset{a}{\overset{x}{\int }}\underset{a}{\overset{{\sigma }_1}{\int }}{({\sigma }_1-t)}^{n-1}f(t)\mathrm{d}t\mathrm{d}{\sigma }_1\\ & =\frac{1}{(n-1)!}\underset{a}{\overset{x}{\int }}\underset{t}{\overset{x}{\int }}{({\sigma }_1-t)}^{n-1}f(t)\mathrm{d}{\sigma }_1\mathrm{d}t\\ & =\frac{1}{n!}\underset{a}{\overset{x}{\int }}{(x-t)}^{n}f(t)\mathrm{d}t\end{array}$

The proof follows.

Applications

In fractional calculus, this formula can be used to construct a notion of differintegral, allowing one to differentiate or integrate a fractional number of times. Integrating a fractional number of times with this formula is straightforward; one can use fractional n by interpreting (n-1)! as Γ(n) (see Gamma function).

References

• Gerald B. Folland, Advanced Calculus, p. 193, Prentice Hall (2002). ISBN 0-13-065265-2

External links

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