Download e-book for iPad: An Introduction to Classical Complex Analysis: 1 by Robert B. Burckel

By Robert B. Burckel

ISBN-10: 0080873987

ISBN-13: 9780080873985

ISBN-10: 0121417018

ISBN-13: 9780121417017

ISBN-10: 376430989X

ISBN-13: 9783764309893

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15 # 1 lY(3 -0 dz V W € c\y. 10(ii) may be cited. Q4 A Useful Sufficient Condition for Differentiability If U is an open subset of C and f: U -+ C a function, there are various conditions on f which insure that it is holomorphic in U. ) One such condition is that f have continuous first partial derivatives which satisfy the so-called Cauchy-Riemann equation in U (see p. ) D J = -iDJ There is a slick proof in HOFFMAN [1975],p. 202. 18 below gives a slight extension. 16 Let U be an open subset of @, zo E U , f: U --f 62 a function such that D1f and D2f exist in U and are continuous at zo.

Let y : [0, 13 --f C be a piecewise smooth curve. < t, = 1. Hint: By adding more points t j the left side is not decreased, so we can suppose that all the points of discontinuity of y' are among the t j . Then y(tj) = /:-1 Y' and 4 Y ) = 27=1j:;-l lY% (iii) With y , t o , . , t, as above, set 6 = maxlsjs,(tj - t j - l ) . Show that i f f is a continuousfunction on the range of y and all the points of discontinuity of y' are among the tj, then Hint: Use the hint to (ii) and the fact that 03 Differentiating under the Integral We will use (and prove) as we progress, several successively stronger theorems on differentiating behind the integral sign.

N - k + l)cn(z - ~ ~ ) nV Z-E D(zo, ~ R). n=k Proof: It suffices to prove that f is differentiable in D(zo,R) and satisfies equation (2. I). 5, induce on k. Also, by considering the function F(z) = f ( z zo) in D(0, R), we may assume without loss of generality that zo = 0. 1). Fix w E D(0, R) and choose IwI < r < R. Then for z E D(0, r)\{w} + The expression in square brackets is 0 if n confirms that it equals n-1 (4) (z - w) 2kw"-'z"-*-'. , R and so, since r < R , the sum in ( 5 ) is finite. Therefore, since D(0, r ) is a neighborhood of w, we have only to let z --f w in ( 5 ) to conclude thef'(w) exists and equals g(w), as claimed.

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An Introduction to Classical Complex Analysis: 1 by Robert B. Burckel

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