New Online Book! Handbook of Mathematical Functions (AMS55) Conversion & Calculation Home >> Reference Information Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables (AMS55) Purchase the electronic edition of this book in Adobe PDF format! FIRST | PREVIOUS | NEXT | LAST | CONTENTS | PAGE | ABOUT | MEDIUM | LARGE Below is the OCR-scanned text from this page: ELEMENTARY ANALYTICAL METHODS 11 3.2.3 min. aan bi>b,>b,> . . . >bn 3.2.7 k = l k a k b X > ( \$ u k ) (8 b k ) Hiilder’s Inequality for Sums 1 1 P q If -+-=1, p>l, q>l equality holds if and only if I b k l = c I o k l P - ’ (c=con- stant>O). If p=q=2 we get Cauchy’s Inequality 3.2.9 ~. [5 ukbkI2<& a: 5 bz (equality for ak=cbk, k = l k = l k = l c constant). Hiilder’s Inequality for Integrals 1 1 P q If -+-=l,p>l, q>l 3.2.10 equality holds if and only if IgCx) I =clfcr) 1P-l (c=constant>O). If p=q=2 we get Schwarz’s Inequality 3.2.11 Minkowski’s Inequality for Sums If p>l and U k , b k > O for all k, 3.2.12 equality holds if and only if b k = C a k (c=con- stant>O). Minkowski’s Inequality for Integrals If P>l, 3.2.13 equality holds if and only if g(x)=cf(x) (c=con- stant>O). 3.3. Rules for Differentiation and Integration Derivatives a du - (cu)=c -) c constant ax ax 3.3.1 3.3.2 3.3.3 d du dv - (u+v)=-+- ax ax ax d dv du - (uv)=u -+v - ax ax ax d vau/ax- udvldx d;E (uIv>= V2 3.3.4 3.3.5 Leibniz’s Theorem for Differentiation of an Integral 3.3.7 The page scan image above, and the text in the text box above, are contributions of the National Institute of Standards and Technology that are not subject to copyright in the United States. ©2000 ConvertIt.com, Inc. All rights reserved. Terms of Use.