differentiation under the integral sign

differentiation under the integral sign

The technique of

differentiation

under the integral sign

concerns the interchange of the

operation

of differentiation
with respect to a

parameter

with the operation of integration
over some other variable:

∂

∂

⁡

x

⁢

∫

Ω

f

⁢

(

x

,

ω

)

⁢

𝑑

ω

=

∫

Ω

∂

∂

⁡

x

⁢

f

⁢

(

x

,

ω

)

⁢

𝑑

ω

.

Intuitively, the rule ought to work because
differentiation commutes with finite summation,
and one may conjecture that it can also
commute 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 and
integration is legal.

Formal statements

Theorem 1

(Elementary Calculus version).

Let

f

:

[

a

,

b

]

×

Y

→

R

be a

function

,
with

[

a

,

b

]

being a

closed interval

, and

Y

being a

compact subset

1

1

Assumed to be Jordan-measurable if the

Riemann integral

is to be used.

of

R

n

.
Suppose that both

f

⁢

(

x

,

y

)

and

∂

⁡

f

⁢

(

x

,

y

)

/

∂

⁡

x

are

continuous

in the variables

x

and

y

jointly.
Then

∫

Y

f

(

x

.

y

)

d

y

exists as a

continuously differentiable function

of

x

on

[

a

,

b

]

, with

derivative

d

d

⁢

x

⁢

∫

Y

f

⁢

(

x

,

y

)

⁢

𝑑

y

=

∫

Y

∂

∂

⁡

x

⁢

f

⁢

(

x

,

y

)

⁢

𝑑

y

.

Theorem 1 is the formulation of integration
under the integral sign that usually appears in

elementary

Calculus

texts.
Unfortunately, its

restriction

that

Y

must
be

compact

can be quite severe for applications: e.g.
integrals over

(

-

∞

,

+

∞

)

are not included.
Theorem 2 below addresses this problem and others:

Theorem 2

(Measure theory version).

Let

X

be an open subset of

R

, and

Ω

be a

measure space

.
Suppose

f

:

X

×

Ω

→

R

satisfies

the following conditions:

1.

f

⁢

(

x

,

ω

)

is a Lebesgue-integrable function of

ω

for each

x

∈

X

.

2.

For almost all

ω

∈

Ω

, the derivative

∂

⁡

f

⁢

(

x

,

ω

)

/

∂

⁡

x

exists for all

x

∈

X

.

3.

There is an integrable function

Θ

:

Ω

→

ℝ

such that

|

∂

⁡

f

⁢

(

x

,

ω

)

/

∂

⁡

x

|

≤

Θ

⁢

(

ω

)

for all

x

∈

X

.

Then for all

x

∈

X

,

d

d

⁢

x

⁢

∫

Ω

f

⁢

(

x

,

ω

)

⁢

𝑑

ω

=

∫

Ω

∂

∂

⁡

x

⁢

f

⁢

(

x

,

ω

)

⁢

𝑑

ω

.

Theorem 2 suffices for many applications,
but using the

Fundamental Theorem of Calculus for Lebesgue integration

,
we can weaken the hypotheses for differentiating under
the integral sign even further:

Theorem 3

.

Let

X

be an open subset of

R

, and

Ω

be a measure space.
Suppose that a function

f

:

X

×

Ω

→

R

satisfies
the following conditions:

1.

f

⁢

(

x

,

ω

)

is a

measurable function

of

x

and

ω

jointly, and is

integrable

over

ω

, for almost all

x

∈

X

held fixed.

2.

For almost all

ω

∈

Ω

,

f

⁢

(

x

,

ω

)

is an

absolutely continuous
function

of

x

. (This guarantees that

∂

⁡

f

⁢

(

x

,

ω

)

/

∂

⁡

x

exists
almost everywhere.)

3.

∂

⁡

f

/

∂

⁡

x

is “locally integrable” — that is, for
all compact intervals

[

a

,

b

]

contained

in

X

:

∫

a

b

∫

Ω

|

∂

∂

⁡

x

⁢

f

⁢

(

x

,

ω

)

|

⁢

𝑑

ω

⁢

𝑑

x

<

∞

.

Then

∫

Ω

f

⁢

(

x

,

ω

)

⁢

𝑑

ω

is an absolutely continuous function of

x

, and for almost every

x

∈

X

, its derivative exists
and is given by

d

d

⁢

x

⁢

∫

Ω

f

⁢

(

x

,

ω

)

⁢

𝑑

ω

=

∫

Ω

∂

∂

⁡

x

⁢

f

⁢

(

x

,

ω

)

⁢

𝑑

ω

.

If the

Kurzweil-Henstock integral

— which has
a stronger

Fundamental Theorem of Calculus

(

http://planetmath.org/FundamentalTheoremOfCalculusForKurzweilHenstockIntegral

)
—
is used in place
of the

Lebesgue integral

,
Theorem 3 can be generalized to a formulation
that provides also the

necessary

conditions 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 integrand
becomes

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 under
the 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

R

m

, and

Ω

be a measure space.
Given

f

⁢

(

x

,

ω

)

, for each

ω

∈

Ω

, a generalized function of

x

∈

X

(in the sense of Schwartz’s theory of

distributions

),
define:

⟨

∫

Ω

f

⁢

(

⋅

,

ω

)

⁢

𝑑

ω

,

ϕ

⟩

:=

∫

Ω

⟨

f

⁢

(

⋅

,

ω

)

,

ϕ

⟩

⁢

𝑑

ω

,

ϕ

∈

𝒟

⁢

(

X

)

.

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

∂

∂

⁡

x

i

⁢

∫

Ω

f

⁢

(

x

,

ω

)

⁢

𝑑

ω

=

∫

Ω

∂

∂

⁡

x

i

⁢

f

⁢

(

x

,

ω

)

⁢

𝑑

ω

.

where

∂

/

∂

⁡

x

i

refers to the generalized derivative
of 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 where
the integrand is not

absolutely continuous

— and so
has a generalized derivative different from the ordinary derivative
— yet its integral has a classical derivative that is
represented 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-integrable
function (of the parameter

x

).

Theorem 4 is not so well-publicized,
but appears, for example, in

[

Jones

]

, and hinted
at 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

α

:

[

a

,

b

]

→

ℝ

,

β

:

[

a

,

b

]

→

ℝ

, and

f

:

[

a

,

b

]

×

ℝ

→

ℝ

, we have:

d

d

⁢

x

⁢

∫

α

⁢

(

x

)

β

⁢

(

x

)

f

⁢

(

x

,

y

)

⁢

𝑑

y

=

∫

α

⁢

(

x

)

β

⁢

(

x

)

∂

⁡

f

⁢

(

x

,

y

)

∂

⁡

x

⁢

𝑑

y

+

d

⁢

β

⁢

(

x

)

d

⁢

x

⁢

f

⁢

(

x

,

β

⁢

(

x

)

)

-

d

⁢

α

⁢

(

x

)

d

⁢

x

⁢

f

⁢

(

x

,

α

⁢

(

x

)

)

.

This result can be extrapolated from Theorem

1

,
with the help of the Fundamental Theorem of Calculus and
the 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 obtained
by 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 under
integrals 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 as
Lebesgue 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.

References

Flanders

Harley Flanders. “Differentiation under the Integral Sign”.

American Mathematical Monthly

, vol. 80 (June-July 1973), p. 615-627.

Folland

Gerald B. Folland.

Real Analysis

: Modern Techniques and Their Applications

, second ed. Wiley-Interscience, 1999.

Jones

D. S. Jones.

The Theory of Generalized Functions

, second ed.
Cambridge University Press, 1982.

Munkres

James R. Munkres.

Analysis on Manifolds

.
Westview Press, 1991.

Schwartz

Laurent Schwartz.

Théorie des Distributions

, vol. I.
Hermann, 1957.

Talvila

Erik Talvila. “

http://www.math.ualberta.ca/ etalvila/papers/difffinal.pdf

Necessary and Sufficient Conditions for Differentiating Under the Integral Sign”.

American Mathematical Monthly

, vol. 108 (June-July 2001), p. 544-548.

The author of this entry has also written an exposition,
“

http://gold-saucer.afraid.org/math/diff-int/diff-int.pdf

Differentiation under the Integral Sign using

Weak Derivatives

”,
containing a proof of Theorem 4 along with detailed computational examples.

Title

differentiation under the integral sign

Canonical name

DifferentiationUnderTheIntegralSign

Date of creation

2013-03-22 16:26:37

Last modified on

2013-03-22 16:26:37

Owner

stevecheng (10074)

Last modified by

stevecheng (10074)

Numerical id

16

Author

stevecheng (10074)

Entry type

Topic

Classification

msc 46F10

Classification

msc 28A25

Classification

msc 26B15

Classification

msc 26A24

Synonym

Leibniz’s rule

Related topic

DerivativeOfRiemannIntegral

Related topic

IntegrationUnderIntegralSign

Related topic

HolomorphicFunctionAssociatedWithContinuousFunction

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