TRENCH_IMPROPER_FUNCTIONS - FUNCTIONS DEFINED BY IMPROPER INTEGRALS William F Trench Professor Emeritus Department of Mathematics Trinity University San

TRENCH_IMPROPER_FUNCTIONS - FUNCTIONS DEFINED BY IMPROPER...

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FUNCTIONS DEFINED BY IMPROPER INTEGRALS William F. Trench Professor Emeritus Department of Mathematics Trinity University San Antonio, Texas, USA [email protected] ©Copyright May 2012, William F. Trench This is a supplement to the author’s Introduction to Real Analysis Reproduction and online posting are permitted for any valid noncommercial edu- cational, mathematical, or scientific purpose. However, charges for profit beyond reasonable printing costs are strictly prohibited. A complete instructor’s solution manual is available on request by email to the author, subject to verification of the requestor’s faculty status.
1 Foreword This is a revised version of Section 7.5 of my Advanced Calculus (Harper & Row, 1978). It is a supplement to my textbook Introduction to Real Analysis , which is refer- enced several times here. You should review Section 3.4 (Improper Integrals) of that book before reading this document. 2 Introduction In Section 7.2 (pp. 462–484) we considered functions of the form F.y/ D Z b a f .x; y/ dx; c DC4 y DC4 d: We saw that if f is continuous on OEa; bŁ STX OEc; dŁ , then F is continuous on OEc; dŁ (Exer- cise 7.2.3, p. 481) and that we can reverse the order of integration in Z d c F.y/ dy D Z d c Z b a f .x; y/ dx ! dy to evaluate it as Z d c F.y/ dy D Z b a Z d c f .x; y/ dy ! dx (Corollary 7.2.3, p. 466). Here is another important property of F . Theorem 1 If f and f y are continuous on OEa; bŁ STX OEc; dŁ; then F.y/ D Z b a f .x; y/ dx; c DC4 y DC4 d; (1) is continuously differentiable on OEc; dŁ and F 0 .y/ can be obtained by differentiating ( 1 ) under the integral sign with respect to y I that is, F 0 .y/ D Z b a f y .x; y/ dx; c DC4 y DC4 d: (2) Here F 0 .a/ and f y .x; a/ are derivatives from the right and F 0 .b/ and f y .x; b/ are derivatives from the left : Proof If y and y C SOHy are in OEc; dŁ and SOHy ¤ 0 , then F.y C SOHy/ NUL F.y/ SOHy D Z b a f .x; y C SOHy/ NUL f .x; y/ SOHy dx: (3) From the mean value theorem (Theorem 2.3.11, p. 83), if x 2 OEa; bŁ and y , y C SOHy 2 OEc; dŁ , there is a y.x/ between y and y C SOHy such that f .x; y C SOHy/ NUL f .x; y/ D f y .x; y/SOHy D f y .x; y.x//SOHy C .f y .x; y.x/ NUL f y .x; y//SOHy: 2
From this and ( 3 ), ˇ ˇ ˇ ˇ ˇ F.y C SOHy/ NUL F.y/ SOHy NUL Z b a f y .x; y/ dx ˇ ˇ ˇ ˇ ˇ DC4 Z b a j f y .x; y.x// NUL f y .x; y/ j dx: (4) Now suppose SI > 0 . Since f y is uniformly continuous on the compact set OEa; bŁ STX OEc; dŁ (Corollary 5.2.14, p. 314) and y.x/ is between y and y C SOHy , there is a ı > 0 such that if j SOH j < ı then j f y .x; y/ NUL f y .x; y.x// j < SI; .x; y/ 2 OEa; bŁ STX OEc; dŁ: This and ( 4 ) imply that ˇ ˇ ˇ ˇ ˇ F.y C SOHy NUL F.y// SOHy NUL Z b a f y .x; y/ dx ˇ ˇ ˇ ˇ ˇ < SI.b NUL a/ if y and y C SOHy are in OEc; dŁ and 0 < j SOHy j < ı . This implies ( 2 ). Since the integral in ( 2 ) is continuous on OEc; dŁ (Exercise 7.2.3, p. 481, with f replaced by f y ), F 0 is continuous on OEc; dŁ .

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