This preview shows pages 1–3. Sign up to view the full content.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
Unformatted text preview: with respect to z . Proof: We know from Thm. 10 that we have a contour integral representation of f at any point z U : f ( z ) = 1 2 i contintegraldisplay C f ( z ) z z dz. (758) Choose a point z U . The derivative of f at z is given by f ( z ) = lim z f ( z + z ) f ( z ) z , (759) if the limit exists. With (758) we have f ( z + z ) f ( z ) z = 1 2 i z contintegraldisplay C f ( z ) z ( z + z ) dz contintegraldisplay C f ( z ) z z dz = 1 2 i z contintegraldisplay C ( z z ) f ( z ) ( z z z ) f ( z ) ( z z z )( z z ) dz = 1 2 i contintegraldisplay C f ( z ) ( z z z )( z z ) dz. (760) Now we compute contintegraldisplay C f ( z ) ( z z z )( z z ) dz contintegraldisplay C f ( z ) ( z z ) 2 dz = contintegraldisplay C zf ( z ) ( z z z )( z z ) 2 dz. (761) We now show that this integral approaches 0 as z 0. f is continuous on C , and therefore bounded: K > 0 :  f ( z )  K z C . Let d be the miminum distance from z to any point on C : d := min z C  z z  1  z z  2 1 d 2 z C (762) Furthermore, with the triangle inequality, d  z z   z z z  +  z  , z C. (763) 128 For  z  d/ 2 we have d 2 d  z   z z z  1  z z z  2 d , z C. (764) Let L := L ( C ) denote the arc length of the curve C . For  z  d/ 2, we use the MLinequality to obtain vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle contintegraldisplay C zf ( z ) ( z z z )( z z ) 2 dz vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle vextendsingle  z  K 2 d 1 d 2 = 2 K d 3  z  , z . (765) This proves that f ( z ) exists, and is given by f ( z ) = 1 2 i contintegraldisplay C f ( z ) ( z z ) 2 dz. (766) Therefore f may be represented at z as a contour integral. This is true for any z U . We repeat the previous argument with f replaced by f and (758) replaced by (766) to obtain a contour integral representation of...
View Full
Document
 Spring '08
 Staff
 Math, Derivative

Click to edit the document details