3.2 General integral depending on a parameter
General integral depending on a parameter.
- Integral on infinite interval.
- Unbounded function (improper integral).
What we’re interested are integrals like

Definition 3.2.1
-
, such that , such that ,i.e.
is pointwisely convergent with respect to for any . -
It is uniformly convergent with respect to
, mentioned above has nothing to do with , i.e. , such that , , such that
Continuity. Consider
For general integrals, we cannot avoid discussing its convergence.
Theorem 3.2.2
Assuming
is uniformly convergent on with respect to .-
, is continuous at with respect to , and is uniformly continuous on any bounded closed interval with respect to , i.e. , such that and ,
Then
Derivability.
Theorem 3.2.3
Given
is convergent. , is continuous with respect to and is uniformly convergent with respect to .
Then in a certain neighborhood

Here the existence of

Since the existence of the first term is unknown, we consider
Notice that
Therefore, we have

where

i.e.
![∫ ∫ ∫ ∫
+∞ +∞ t +∞ -∂f-
f(x,y0 + tek)dx = f (x,y0)dx + ∖lef t[ ∂y (x,y0 + sek)dx ∖right ]ds
a a 0 a k](main168x.png)
and it’s uniformly convergent with respect to

When

The selection of
Integrability.
Theorem 3.2.4
Assuming
