InterchangeDiffandIntegral

InterchangeDiffandIntegral - 2005-06 Second Term...

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Unformatted text preview: 2005-06 Second Term MAT2060B1Supplementary Notes 3Interchange of Differentiation and IntegrationThe theme of this course is about various limiting processes. We have learntthe limits of sequences of numbers and functions, continuity of functions, limits ofdifference quotients (derivatives), and even integrals are limits of Riemann sums.As often encountered in applications, exchangeability of limiting processes is animportant topic. For example, we learntddxZxaf=Zxadfdx,f(a) = 0,wheneverdfdxis integrable; alsolimnddxfn(x) =ddxlimnfn(x),if{fn}and{fn}converge uniformly.Here we consider the following situation. Letf(x, y) be a function defined in[a, b][c, d] and(y) =Zbaf(x, y)dx.It is natural to ask if continuity and differentiability are preserved under integration.Theorem 1.Letf(x, y)be continuous in[a, b][c, d]. Thendefined above is acontinuous function on[c, d].Proof.Sincefis continuous in [a, b][c, d], it is bounded and uniformly continuous.In other words, for any >0,such that|(y)(y)| Zba|f(x, y)f(x, y)|dx< (ba)y,|yy|< ,which shows thatis uniformly continuous on [c, d].Source: http://www.math.cuhk.edu.hk/~mat2060/mat2060b/Notes/notes3.pdf2005-06 Second Term MAT2060B2Theorem 2.Letfandfybe continuous in[a, b][c, d]. Thenis differentiableandddy(y) =Zbafy(x, y)dxholds.Proof.Fixy(c, d),y+h(c, d) for smallhR,(y+h)(y)h=1hZba(f(x, y+h)f(x, y))dx=Zbafy(x, z)dxwherezis a point betweenyandy+hwhich depends onx. In any case,(y+h)(y)hZbafy(x, y)dxZbafy(x, z)fy(x, y)dx.Sincefyis uniformly continuous on [a, b][c, d], for >0,such thatfy(x, y)fy(x, y)< ,|yy|< andx.Takingh, we get(y+h)(y)hZbafy(x, y)dx< ,whence the condition follows.Wheny=cord, the same proof works with some trivial changes.In many applications, the rectangle is replaced by an unbounded region. Whenthis happens, we need to consider improper integrals. As a typical case, lets assumefis defined in [a,)[c, d] and set(y) =Zaf(x, y)dx.2005-06 Second Term MAT2060B3The function(y) makes sense if the improper integralZaf(x)dxis well-definedfor eachy. Recall that this meanslimbZbaf(x, y)dxexists. We introduce the following definition: The improper integralZaf(x, y)dxis uniformly convergent if,b>0 such thatZbbf(x, y)dx< ,b, bb....
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InterchangeDiffandIntegral - 2005-06 Second Term...

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