Complex Analysis (Graduate Texts in Mathematics, Volume 103) by Serge Lang

By Serge Lang

Now in its fourth version, the 1st a part of this publication is dedicated to the elemental fabric of advanced research, whereas the second one covers many certain subject matters, comparable to the Riemann Mapping Theorem, the gamma functionality, and analytic continuation. energy sequence tools are used extra systematically than is located in different texts, and the ensuing proofs usually shed extra gentle at the effects than the normal proofs. whereas the 1st half is appropriate for an introductory direction at undergraduate point, the extra issues lined within the moment half provide the teacher of a gradute direction loads of flexibility in structuring a extra complicated direction.

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Let the new coordinates u1 ; u2 ; u3 be de®ned by specifying the Cartesian coordinates (x1 ; x2 ; x3 ) as functions of (u1 ; u2 ; u3 †: x1 ˆ f …u1 ; u2 ; u3 †; x2 ˆ g…u1 ; u2 ; u3 †; x3 ˆ h…u1 ; u2 ; u3 †; …1:54† where f, g, h are assumed to be continuous, di€erentiable. A point P (Fig. 16) in space can then be de®ned not only by the rectangular coordinates (x1 ; x2 ; x3 ) but also by curvilinear coordinates (u1 ; u2 ; u3 ). If u2 and u3 are constant as u1 varies, P (or its position vector r) describes a curve which we call the u1 coordinate curve.

Alternatively dV can be written as     @…x1 ; x2 ; x3 †  @r @r @r   du du du ;  Á  du du du ˆ dV ˆ  @u1 @u2 @u3  1 2 3  @…u1 ; u2 ; u3 †  1 2 3 where   @x1   @u1   @…x1 ; x2 ; x3 †  @x2 ˆ Jˆ @…u1 ; u2 ; u3 †  @u1   @x3   @u1 @x1 @u2 @x2 @u2 @x3 @u2 …1:57†  @x1   @u3   @x2   @u3   @x3   @u3  is called the Jacobian of the transformation. 54) is one to one in the neighborhood of a point. We are now ready to express the gradient, divergence, and curl in terms of u1 ; u2 , and u3 .

Tensors of second rank From two contravariant vectors A and B we may form the N 2 quantities A B . This is known as the outer product of tensors. These N 2 quantities form the components of a contravariant tensor of the second rank: any aggregate of N 2 quantities T  which, under a coordinate change, transform like the product of 48 BASIC OPERATIONS WITH TE NS ORS two contravariant vectors T  ˆ @x @x H T ; @x H @x H …1:97† is a contravariant tensor of rank two. We may also form a covariant tensor of rank two from two covariant vectors, which transforms according to the formula T ˆ @x H @x H H T : @x @x …1:98† Similarly, we can form a mixed tensor T   of order two that transforms as follows: T  ˆ @x @x H H T : @x H @x …1:99† We may continue this process and multiply more than two vectors together, taking care that their indexes are all di€erent.

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