differentiation under the integral sign

The technique of differentiation under the integral signconcerns the interchange of the operation of differentiationwith respect to a parameter with the operation of integrationover some other variable:

$$\frac{\partial }{\partial x}{\int }_{\mathrm{\Omega }}f(x,\omega )𝑑\omega ={\int }_{\mathrm{\Omega }}\frac{\partial }{\partial x}f(x,\omega )𝑑\omega .$$

Intuitively, the rule ought to work becausedifferentiation commutes with finite summation,and one may conjecture that it can alsocommute with infinite summation (in the form of the integral),at least in some cases.

The theorems below give some sufficient conditions,in increasing generality and sophistication,for which the swap of differentiation andintegration is legal.

Formal statements

Theorem 1 (Elementary Calculus version).

Let $f\mathrm{:}\mathrm{[}a\mathrm{,}b\mathrm{]}\mathrm{\times }Y\mathrm{\to }\mathrm{R}$ be a function,with $\mathrm{[}a\mathrm{,}b\mathrm{]}$ being a closed interval, and$Y$ being a compact subset of ${\mathrm{R}}^n$.Suppose that both $f\mathit{}\mathrm{(}x\mathrm{,}y\mathrm{)}$ and $\mathrm{\partial }\mathit{}f\mathit{}\mathrm{(}x\mathrm{,}y\mathrm{)}\mathrm{/}\mathrm{\partial }\mathit{}x$ are continuousin the variables $x$ and $y$ jointly.Then ${\mathrm{\int }}_{Y}f\mathrm{(}x\mathrm{.}y\mathrm{)}dy$ exists as a continuously differentiable function of$x$ on $\mathrm{[}a\mathrm{,}b\mathrm{]}$, with derivative

$$\frac{d}{dx}{\int }_{Y}f(x,y)𝑑y={\int }_Y\frac{\partial }{\partial x}f(x,y)𝑑y.$$

Theorem 1 is the formulation of integrationunder the integral sign that usually appears in elementary Calculus texts.Unfortunately, its restriction that $Y$ mustbe compact can be quite severe for applications: e.g.integrals over $(-\mathrm{\infty },+\mathrm{\infty })$ are not included.Theorem 2 below addresses this problem and others:

Theorem 2 (Measure theory version).

Let $X$ be an open subset of $\mathrm{R}$, and $\mathrm{\Omega }$ be a measure space.Suppose $f\mathrm{:}X\mathrm{\times }\mathrm{\Omega }\mathrm{\to }\mathrm{R}$ satisfiesthe following conditions:

1. $f(x,\omega )$ is a Lebesgue-integrable function of $\omega$ for each $x\in X$.

2. For almost all$\omega \in \mathrm{\Omega }$, the derivative $\partial f(x,\omega )/\partial x$ exists for all $x\in X$.

3. There is an integrable function $\mathrm{\Theta }:\mathrm{\Omega }\to ℝ$ such that$\left|\partial f(x,\omega )/\partial x\right|\le \mathrm{\Theta }(\omega )$for all $x\in X$.

Then for all $x\mathrm{\in }X$,

$$\frac{d}{dx}{\int }_{\mathrm{\Omega }}f(x,\omega )𝑑\omega ={\int }_{\mathrm{\Omega }}\frac{\partial }{\partial x}f(x,\omega )𝑑\omega .$$

Theorem 2 suffices for many applications,but using the Fundamental Theorem of Calculus for Lebesgue integration,we can weaken the hypotheses for differentiating underthe integral sign even further:

Theorem 3.

Let $X$ be an open subset of $\mathrm{R}$, and $\mathrm{\Omega }$ be a measure space.Suppose that a function $f\mathrm{:}X\mathrm{\times }\mathrm{\Omega }\mathrm{\to }\mathrm{R}$ satisfiesthe following conditions:

1. $f(x,\omega )$ is a measurable function of $x$ and $\omega$jointly, and is integrable over $\omega$, for almost all $x\in X$ held fixed.

2. For almost all$\omega \in \mathrm{\Omega }$, $f(x,\omega )$ is an absolutely continuousfunction of $x$. (This guarantees that $\partial f(x,\omega )/\partial x$ existsalmost everywhere.)

3. $\partial f/\partial x$ is “locally integrable” — that is, forall compact intervals $[a,b]$ contained in $X$:

Then ${\mathrm{\int }}_{\mathrm{\Omega }}f\mathit{}\mathrm{(}x\mathrm{,}\omega \mathrm{)}\mathit{}𝑑\omega$ is an absolutely continuous function of $x$, and for almost every $x\mathrm{\in }X$, its derivative existsand is given by

$$\frac{d}{dx}{\int }_{\mathrm{\Omega }}f(x,\omega )𝑑\omega ={\int }_{\mathrm{\Omega }}\frac{\partial }{\partial x}f(x,\omega )𝑑\omega .$$

If the Kurzweil-Henstock integral — which hasa stronger Fundamental Theorem of Calculus (http://planetmath.org/FundamentalTheoremOfCalculusForKurzweilHenstockIntegral)—is used in placeof the Lebesgue integral,Theorem 3 can be generalized to a formulationthat provides also the necessaryconditions for differentiation under the integral sign.See [Talvila] for the full details.

Yet this is not the end of the story.There are some applications in which the integrand is too “irregular”,or the integral of the differentiated integrandbecomes divergent, and neither Theorem 2 or Theorem 3 would apply.However, if we use generalized functions(all of which can be differentiated at will),then we can extend the technique of differentiation underthe integral sign further,and make sense of any “irregular” integrals that may result:

Theorem 4 (Distribution theory version).

Let $X$ be an open set in ${\mathrm{R}}^m$, and $\mathrm{\Omega }$ be a measure space.Given $f\mathit{}\mathrm{(}x\mathrm{,}\omega \mathrm{)}$, for each $\omega \mathrm{\in }\mathrm{\Omega }$, a generalized function of $x\mathrm{\in }X$ (in the sense of Schwartz’s theory of distributions),define:

$$⟨{\int }_{\mathrm{\Omega }}f(\cdot ,\omega )𝑑\omega ,\varphi ⟩:={\int }_{\mathrm{\Omega }}⟨f(\cdot ,\omega ),\varphi ⟩𝑑\omega ,\varphi \in 𝒟(X).$$

Assumethe above integral is well-defined and gives a distribution.Then

$$\frac{\partial }{\partial x_i}{\int }_{\mathrm{\Omega }}f(x,\omega )𝑑\omega ={\int }_{\mathrm{\Omega }}\frac{\partial }{\partial x_i}f(x,\omega )𝑑\omega .$$

where $\mathrm{\partial }\mathrm{/}\mathrm{\partial }\mathit{}{x}_i$ refers to the generalized derivativeof generalized functions on both sides of the equation.

For an absolutely continuous function,the generalized derivative coincides with the ordinary derivative,so Theorem 4 indeed generalizes Theorem 3.On the other hand, there are cases wherethe integrand is not absolutely continuous — and sohas a generalized derivative different from the ordinary derivative— yet its integral has a classical derivative that isrepresented by the final equation of Theorem 4. For instance,the integrand may involve a step function,and its derivative would thus involve a Dirac delta distribution,that when integrated, yields an ordinary locally-integrablefunction (of the parameter $x$).

Theorem 4 is not so well-publicized,but appears, for example, in [Jones], and hintedat in a comment in [Schwartz].

Other variations

There are other frequently-used variations of the theorems above.

Moving domains of integration.Not only can the integrand vary with the parameter,we can consider domains of integration,subsets of ${ℝ}^n$, that vary with the parameter.

In the one-dimensional case,for continuously differentiable functions$\alpha :[a,b]\to ℝ$, $\beta :[a,b]\to ℝ$, and$f:[a,b]\times ℝ\to ℝ$, we have:

$$\frac{d}{dx}{\int }_{\alpha (x)}^{\beta (x)}f(x,y)𝑑y={\int }_{\alpha (x)}^{\beta (x)}\frac{\partial f(x,y)}{\partial x}𝑑y+\frac{d\beta (x)}{dx}f(x,\beta (x))-\frac{d\alpha (x)}{dx}f(x,\alpha (x)).$$

This result can be extrapolated from Theorem 1,with the help of the Fundamental Theorem of Calculus andthe multi-variate chain rule (http://planetmath.org/ChainRuleSeveralVariables).

Generalizations to varying smooth surfaces or volumes — or, more generally,$k$-dimensional differentiable manifolds in ${ℝ}^n$ — can be obtainedby using integrals of differential forms on chains (http://planetmath.org/NChain),and Stokes’ Theorem. Details can be found in [Flanders].

Different types of integrals.The differentiation can also be taken underintegrals other than of the standard Riemann type,such as the line integrals and surface integrals of vector calculus,or complex contour integrals.(Actually, these kinds of integrals can be re-formulated asLebesgue integrals, so Theorem 2 applies to them.)

Complex variables.Other applications require differentiating holomorphic functions with respect to a complex variable, and Theorem 2 generalizes directly to this situation, without requiring differentiation with respect to real variables as an intermediary.

The author of this entry has also written an exposition,“http://gold-saucer.afraid.org/math/diff-int/diff-int.pdfDifferentiation under the Integral Sign using Weak Derivatives”,containing a proof of Theorem 4 along with detailed computational examples.