Parameter integral

As parameters integral in being Analysis an integral referred whose integrand of a parameter dependent. An important example is Euler's representation of the gamma function . The value of such an integral is then a function of the parameter and the question arises, for example, whether this function is continuous or differentiable.

Definition of the parameter integral

There are a measure space, , a Banach space and . For everyone is about integrable in terms of measure . Then is called ${\displaystyle (\Omega ,{\mathcal {A}},\mu )}$ ${\displaystyle \varnothing \neq X\subseteq \mathbb {R} ^{n}}$ ${\displaystyle (E,\Vert \cdot \Vert )}$ ${\displaystyle f\colon X\times \Omega \to E}$ ${\displaystyle x\in X}$ ${\displaystyle \omega \mapsto f(x,\omega )}$ ${\displaystyle \Omega }$ ${\displaystyle \mu }$ ${\displaystyle F\colon X\to E}$

${\displaystyle F(x)=\int _{\Omega }f(x,\omega )\,\mu (\mathrm {d} \omega )}$

Parameter integral with the parameter . ${\displaystyle x}$

Example of parameter integrals

The gamma function

${\displaystyle \Gamma (x)=\int _{0}^{\infty }t^{x-1}e^{-t}\,\mathrm {d} t.}$

Continuity of parameter integrals

Be a metric space, a Banach space, a measure space. For an illustration, apply ${\displaystyle X}$ ${\displaystyle (E,\Vert \cdot \Vert )}$ ${\displaystyle (\Omega ,{\mathcal {A}},\mu )}$ ${\displaystyle f\colon X\times \Omega \to E}$

• ${\displaystyle f(x,\cdot )\in {\mathcal {L}}_{1}(\Omega ,\mu ,E)}$for each , ${\displaystyle x\in X}$
• ${\displaystyle f(\cdot ,\omega )\in C(X,E)}$( i.e. continuously) for -fa , ${\displaystyle \mu }$ ${\displaystyle \omega \in \Omega }$
• There's a with for . ${\displaystyle g\in {\mathcal {L}}_{1}(\Omega ,\mu ,\mathbb {R} )}$ ${\displaystyle \Vert f(x,\omega )\Vert \leqslant g(\omega )}$ ${\displaystyle (x,\omega )\in X\times \Omega }$

Then

${\displaystyle F\colon X\to E,\ x\mapsto \int _{\Omega }f(x,\omega )\mu (\mathrm {d} \omega )}$

well-defined and steady.

Differentiability of parameter integrals

Be open, a Banach space, a measure space. For an illustration, apply ${\displaystyle U\subset \mathbb {R} ^{d}}$ ${\displaystyle (E,\Vert \cdot \Vert )}$ ${\displaystyle (\Omega ,{\mathcal {A}},\mu )}$ ${\displaystyle f\colon U\times \Omega \to E}$

• ${\displaystyle f(u,\cdot )\in {\mathcal {L}}_{1}(\Omega ,\mu ,E)}$for each , ${\displaystyle u\in U}$
• ${\displaystyle f(\cdot ,\omega )\in C^{1}(U,E)}$( i.e. continuously differentiable) for -fa , ${\displaystyle \mu }$ ${\displaystyle \omega \in \Omega }$
• There's a with for . ${\displaystyle g\in {\mathcal {L}}_{1}(\Omega ,\mu ,\mathbb {R} )}$ ${\displaystyle \Vert \partial _{u}f(u,\omega )\Vert \leqslant g(\omega )}$ ${\displaystyle (u,\omega )\in U\times \Omega }$

Then

${\displaystyle F\colon U\to E,\ u\mapsto \int _{\Omega }f(u,\omega )\mu (\mathrm {d} \omega )}$

continuously differentiable with

${\displaystyle \partial _{j}F(u)=\int _{\Omega }{\frac {\partial }{\partial u^{j}}}f(u,\omega )\mu (\mathrm {d} \omega ),\quad u\in U,\quad 1\leqslant j\leqslant d.}$

Note: ${\displaystyle \partial _{u}\int \cdots =\int \partial _{u}\cdots }$

Leibniz rule for parameter integrals

In practice it is also relevant how one derives parameter integrals with functions that are dependent on within the limits. According to the Leibniz rule, this is done using the following procedure: ${\displaystyle \omega }$

For continuously differentiable functions , and holds ${\displaystyle \chi }$ ${\displaystyle \varphi }$ ${\displaystyle f}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \omega }}\int \limits _{\chi (\omega )}^{\varphi (\omega )}f(x,\omega )\mathrm {d} x=\int \limits _{\chi (\omega )}^{\varphi (\omega )}{\frac {\partial }{\partial \omega }}f(x,\omega )\mathrm {d} x+f(\varphi (\omega ),\omega ){\frac {\mathrm {d} }{\mathrm {d} \omega }}\varphi (\omega )-f(\chi (\omega ),\omega ){\frac {\mathrm {d} }{\mathrm {d} \omega }}\chi (\omega ).}$

Derivation

One can derive this rule by using the product and chain rules. In this integral there are three functions that depend on and according to these are derived individually, while the others are retained as long as: ${\displaystyle \omega \,}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \omega }}\int \limits _{\chi (\omega )}^{\varphi (\omega )}f(x,\omega )\mathrm {d} x={\frac {\mathrm {d} }{\mathrm {d} \omega }}\int \limits _{\chi }^{\varphi }f(x,\omega )\mathrm {d} x+{\frac {\mathrm {d} }{\mathrm {d} \omega }}\int \limits _{\chi }^{\varphi (\omega )}f(x)\mathrm {d} x+{\frac {\mathrm {d} }{\mathrm {d} \omega }}\int \limits _{\chi (\omega )}^{\varphi }f(x)\mathrm {d} x}$

How to derive the first term on the right-hand side is given in the rule of thumb for differentiability of parameter integrals .

On the one comes close, by applying the chain rule. This is how the second term on the right becomes: ${\displaystyle {\varphi (\omega )}\,}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} \omega }}\int \limits _{\chi }^{\varphi (\omega )}f(x)\mathrm {d} x={\frac {\mathrm {d} }{\mathrm {d} {\varphi }}}{\Big (}{\int \limits _{\chi }^{\varphi }f(x)\mathrm {d} x}{\Big )}\cdot {\frac {\mathrm {d} {\varphi }}{\mathrm {d} \omega }}}$

A distinction is also made between the variable . ${\displaystyle {\chi (\omega )}\,}$

As a result we get the derivative of the integral after the upper limit multiplied by and secondly one after the lower limit multiplied by . One can actually work out these integrals, as one knows from the fundamental theorem of analysis . ${\displaystyle \varphi '(\omega )}$ ${\displaystyle \chi '(\omega )}$

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} {\varphi }}}{\int \limits _{\chi }^{\varphi }f(x)\mathrm {d} x}={\frac {\mathrm {d} }{\mathrm {d} {\varphi }}}{\big (}F(\varphi )-F(\chi ){\big )}=f(\varphi )}$

and

${\displaystyle {\frac {\mathrm {d} }{\mathrm {d} {\chi }}}{\int \limits _{\chi }^{\varphi }f(x)\mathrm {d} x}={\frac {\mathrm {d} }{\mathrm {d} {\chi }}}{\big (}F(\varphi )-F(\chi ){\big )}=-f(\chi )}$

Everything together then leads to the Leibniz rule for parameter integrals, as it stands above.

literature

• Harro Heuser : Analysis 2nd 9th edition, Teubner, 1995, ISBN 3-51932-232-3 , pp. 101ff.
• René L. Schilling: Measures, Integrals and Martingales 3rd edition, Cambridge University Press, 2011, ISBN 978-0-521-61525-9 , pp. 92ff.