Differentiation under the integral sign

As a parameter will be referred to an integral integral, whose integrant is dependent on a parameter in the analysis. An important example is the representation of the Euler's Gamma function. The value of such an integral is then a function of the parameter and it represents, for example, the question of whether this function is continuous or differentiable.

Definition of the parameter integral

Let be a measure space, a Banach space and. For all was over integrated in the measurement. Then say

Integral parameters with the parameter.

Continuity of parameter integrals

Be a metric space is a Banach space, a measure space. For an illustration applies

• For each,
• (ie continuous) for,
• There is a with for.

Then

Well- defined and continuous.

Differentiability of parameter integrals

Be open, be a Banach space, a measure space. For an illustration applies

• For each,
• (ie continuously differentiable ) for,
• There is a with for.

Then

Continuously differentiable with

Note:

Leibniz rule for integrals with parameters

For the practice is also relevant, how to derive parameter integrals with dependent function of the limits. After the Leibniz rule that happens by the following method:

For continuously differentiable functions, and is

Or in differential notation, according to Leibniz

Derivation

One can easily derive this rule, so to speak, by proceeding as in product as well as the chain rule. In this integral three features that will depend from and after these are individually derived, while the others are held as long as:

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

On the one comes close, by applying the chain rule. Thus, from the second term of the right side:

So is also differentiated by the variable.

As a result, we again obtain the derivative of the integral with respect to the upper limit with multiplied and, secondly, according to the lower limit multiplied by. These integrals can actually figure out how to know from the Fundamental Theorem of Calculus.

And

All together then leads to the Leibniz rule for integrals with parameters, such as the one above.