Complex Analysis and Differential Equations by Luis Barreira & Claudia Valls

Author:Luis Barreira & Claudia Valls [Barreira, Luis & Valls, Claudia]
Language: eng
Format: epub
Tags: Differential Equations, Calculus, Complex Analysis, General, Functional Analysis, Mathematics, Mathematical Analysis
ISBN: 9781447140085
Google: SvLHFBOjoXQC
Publisher: Springer Science & Business Media
Published: 2012-04-23T00:00:00+00:00

Moreover,

It then follows from the Residue theorem that

(4.46)

On the other hand,

and

(4.47)

We also have

when R→+∞, and analogously,

when R→+∞. It then follows from (4.46) and (4.47) that

when R→+∞. We finally obtain

Figure 4.9Path γ=γ 1+γ 2+γ 3+γ 4

Problem 4.25

For each a>0, show that

(4.48)

Solution

Take r,R>0 such that r<a<R. We consider the path γ=γ 1+γ 2+γ 3+γ 4, where

(see Figure 4.5). We also consider the function

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