Smoluchowski coagulation equation: Difference between revisions
en>YFdyh-bot m r2.7.3) (Robot: Adding fr:Équation de coagulation de Smoluchowski |
en>Anrnusna No edit summary |
||
| Line 1: | Line 1: | ||
The '''Cauchy formula for repeated integration''', named after [[Augustin Louis Cauchy]], allows one to compress ''n'' [[antidifferentiation]]s of a function into a single integral (cf. [[Antiderivative#Techniques of integration|Cauchy's formula]]). | |||
==Scalar case== | |||
Let ''ƒ'' be a continuous function on the real line. Then the ''n''th repeated integral of ''ƒ'' based at ''a'', | |||
:<math>f^{(-n)}(x) = \int_a^x \int_a^{\sigma_1} \cdots \int_a^{\sigma_{n-1}} f(\sigma_{n}) \, \mathrm{d}\sigma_{n} \cdots \, \mathrm{d}\sigma_2 \, \mathrm{d}\sigma_1</math>, | |||
is given by single integration | |||
:<math>f^{(-n)}(x) = \frac{1}{(n-1)!} \int_a^x\left(x-t\right)^{n-1} f(t)\,\mathrm{d}t</math>. | |||
A proof is given by [[mathematical induction|induction]]. Since ''ƒ'' is continuous, the base case follows from the [[Fundamental Theorem of Calculus|Fundamental theorem of calculus]]: | |||
:<math>\frac{\mathrm{d}}{\mathrm{d}x} f^{(-1)}(x) = \frac{\mathrm{d}}{\mathrm{d}x}\int_a^x f(t)\,\mathrm{d}t = f(x)</math>; | |||
where | |||
:<math>f^{(-1)}(a) = \int_a^a f(t)\,\mathrm{d}t = 0</math>. | |||
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, | |||
:<math> | |||
\begin{align} | |||
f^{-(n+1)}(x) &= \int_a^x \int_a^{\sigma_1} \cdots \int_a^{\sigma_{n}} f(\sigma_{n+1}) \, \mathrm{d}\sigma_{n+1} \cdots \, \mathrm{d}\sigma_2 \, \mathrm{d}\sigma_1 \\ | |||
&= \frac{1}{(n-1)!} \int_a^x \int_a^{\sigma_1}\left(\sigma_1-t\right)^{n-1} f(t)\,\mathrm{d}t\,\mathrm{d}\sigma_1 \\ | |||
&= \frac{1}{(n-1)!} \int_a^x \int_t^x\left(\sigma_1-t\right)^{n-1} f(t)\,\mathrm{d}\sigma_1\,\mathrm{d}t \\ | |||
&= \frac{1}{n!} \int_a^x \left(x-t\right)^n f(t)\,\mathrm{d}t | |||
\end{align} | |||
</math> | |||
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== | |||
*{{cite web|author=Alan Beardon|url=http://nrich.maths.org/public/viewer.php?obj_id=1369|title=Fractional calculus II|publisher=University of Cambridge|year=2000}} | |||
[[Category:Integral calculus]] | |||
[[Category:Theorems in analysis]] | |||
Revision as of 05:33, 23 September 2013
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,
is given by single integration
A proof is given by induction. Since ƒ is continuous, the base case follows from the Fundamental theorem of calculus:
where
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,
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