An introduction to the study of the elements of the differential and integral calculus. From the German of the late Axel Harnack, With the permission of the author.

~ 151, 152.

Differentiation with respect to a parameter.

267

a determinate value of oc an interval can be assigned such that
a(, a) within the domain from a to ac + h and from x = a to x = b
is generally a continuous function of both variables, we' have:
b b
a ff(x a)dx= -=f (- dx.
a a
Moreover also it is sufficient for the validity of the theorem,
that the definite integral be such a function of its upper limit x and
of the parameter a as to have its first derived functions with respect
to x and a continuous and the order of differentiations with respect
to x and a interchangeable (~ 54). For, writing:
x
Jf (x, a) dx == F(x, a),
a
we have then:
OF O' f(x, a) ___ d2 x
3x '(zX a), a), x x a= axd a (Xa)x,
and from the equation:
x
af(x,a) a 2 r
8c~ - x f(x, Oc)dx
a
follows by integration:
d X
If -~, ataa d x (X, a)dx.
a a
When the limits of the integral likewise depend on the parameter a,
we obtain its derived function with respect to a, on the hypothesis that
differentiation is admissible under the integral sign, by the formula:
b b,; f(X) a) - f(b, a ) da -f(a, ) da.
a a
152. The preceding theorems require to be supplemented, when
the limits of the integral or the functions to be integrated become
infinite. Without exhausting all possibilities, we consider the following
cases.
a) Although f(x, a) be a continuous function of both variables
in the domain from x = a to x = oc and from a = fp to a = y, still
it is only on a certain hypothesis that the integral:
0o
f f(x, )c)dx
a