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1.1: Introduction to Improper Functions

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In Section 7.2 (pp. 462–484) we considered functions of the form

F(y)=baf(x,y)dx,cyd.

We saw that if f is continuous on [a,b]×[c,d], then F is continuous on [c,d] (Exercise 7.2.3, p. 481) and that we can reverse the order of integration in

dcF(y)dy=dc(baf(x,y)dx)dy

to evaluate it as

dcF(y)dy=ba(dcf(x,y)dy)dx

(Corollary 7.2.3, p. 466).

Here is another important property of F.

[theorem:1] If f and fy are continuous on [a,b]×[c,d], then

F(y)=baf(x,y)dx,cyd,

is continuously differentiable on [c,d] and F(y) can be obtained by differentiating [eq:1] under the integral sign with respect to y; that is,

F(y)=bafy(x,y)dx,cyd.

Here F(a) and fy(x,a) are derivatives from the right and F(b) and fy(x,b) are derivatives from the left.

If y and y+Δy are in [c,d] and Δy0, then

F(y+Δy)F(y)Δy=baf(x,y+Δy)f(x,y)Δydx.

From the mean value theorem (Theorem 2.3.11, p. 83), if x[a,b] and y, y+Δy[c,d], there is a y(x) between y and y+Δy such that

f(x,y+Δy)f(x,y)=fy(x,y)Δy=fy(x,y(x))Δy+(fy(x,y(x)fy(x,y))Δy.

From this and [eq:3],

|F(y+Δy)F(y)Δybafy(x,y)dx|ba|fy(x,y(x))fy(x,y)|dx.

Now suppose ϵ>0. Since fy is uniformly continuous on the compact set [a,b]×[c,d] (Corollary 5.2.14, p. 314) and y(x) is between y and y+Δy, there is a δ>0 such that if |Δ|<δ then

|fy(x,y)fy(x,y(x))|<ϵ,(x,y)[a,b]×[c,d].

This and [eq:4] imply that

|F(y+ΔyF(y))Δybafy(x,y)dx|<ϵ(ba)

if y and y+Δy are in [c,d] and 0<|Δy|<δ. This implies [eq:2]. Since the integral in [eq:2] is continuous on [c,d] (Exercise 7.2.3, p. 481, with f replaced by fy), F is continuous on [c,d].

[example:1] Since

f(x,y)=cosxy\quad and\quadfy(x,y)=xsinxy

are continuous for all (x,y), Theorem [theorem:1] implies that if

F(y)=π0cosxydx,<y<,

then

F(y)=π0xsinxydx,<y<.

(In applying Theorem [theorem:1] for a specific value of y, we take R=[0,π]×[ρ,ρ], where ρ>|y|.) This provides a convenient way to evaluate the integral in [eq:6]: integrating the right side of [eq:5] with respect to x yields

F(y)=sinxyy|πx=0=sinπyy,y0.

Differentiating this and using [eq:6] yields

π0xsinxydx=sinπyy2πcosπyy,y0.

To verify this, use integration by parts.

We will study the continuity, differentiability, and integrability of

F(y)=baf(x,y)dx,yS,

where S is an interval or a union of intervals, and F is a convergent improper integral for each yS. If the domain of f is [a,b)×S where <a<b, we say that F is pointwise convergent on S or simply convergent on S, and write

baf(x,y)dx=limrbraf(x,y)dx

if, for each yS and every ϵ>0, there is an r=r0(y) (which also depends on ϵ) such that

|F(y)raf(x,y)dx|=|brf(x,y)dx|<ϵ,r0(y)y<b.

If the domain of f is (a,b]×S where a<b<, we replace [eq:7] by

baf(x,y)dx=limra+brf(x,y)dx

and [eq:8] by

|F(y)brf(x,y)dx|=|raf(x,y)dx|<ϵ,a<rr0(y).

In general, pointwise convergence of F for all yS does not imply that F is continuous or integrable on [c,d], and the additional assumptions that fy is continuous and bafy(x,y)dx converges do not imply [eq:2].

[example:2] The function

f(x,y)=ye|y|x

is continuous on [0,)×(,) and

F(y)=0f(x,y)dx=0ye|y|xdx

converges for all y, with

F(y)={1y<0,0y=0,1y>0;

therefore, F is discontinuous at y=0.

[example:3] The function

f(x,y)=y3ey2x

is continuous on [0,)×(,). Let

F(y)=0f(x,y)dx=0y3ey2xdx=y,<y<.

Then

F(y)=1,<y<.

However,

0y(y3ey2x)dx=0(3y22y4x)ey2xdx={1,y0,0,y=0,

so

F(y)0f(x,y)ydx\quad if\quady=0.


This page titled 1.1: Introduction to Improper Functions is shared under a CC BY-NC-SA 3.0 license and was authored, remixed, and/or curated by William F. Trench via source content that was edited to the style and standards of the LibreTexts platform.

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