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-R Use lemma 3.2.9 to prove that Jot R 1 z² +4 CR R dz0 as R→∞.

-R Use lemma 3.2.9 to prove that Jot R 1 z² +4 CR R dz0 as R→∞.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Lemma 3.2.9 (M-L lemma). Let I be a regular curve in C, and let f: I → C be continuous. Then V f(z) dz ≤ max|ƒ(z)|l(T). ΖΕΓ Proof. We simply calculate, using lemma 3.1.2, that t1 rt₁ t1 |y'(t)\ | √, f(z) dz| = | ƒ^"^ f(y(t))r' (t) dt| ≤ f*"* |ƒ(~7(1)||/' (t)\ dt < max|ƒ(2) ["* |\ ' (t)\ dt to r to to = max|f(z)|l(I). ΖΕΓ
Transcribed Image Text:Lemma 3.2.9 (M-L lemma). Let I be a regular curve in C, and let f: I → C be continuous. Then V f(z) dz ≤ max|ƒ(z)|l(T). ΖΕΓ Proof. We simply calculate, using lemma 3.1.2, that t1 rt₁ t1 |y'(t)\ | √, f(z) dz| = | ƒ^"^ f(y(t))r' (t) dt| ≤ f*"* |ƒ(~7(1)||/' (t)\ dt < max|ƒ(2) ["* |\ ' (t)\ dt to r to to = max|f(z)|l(I). ΖΕΓ
Question 4. Let R > 0 be a real number and let C denote the semicircular arc of radius R in the upper half of the complex plane centred at the origin from R to - R. - R Use lemma 3.2.9 to prove that Joh 1 2² +4 CH R R dz→0 as R→∞. It is vital to master the technique of evaluating the limiting value of integrals such as this many contour integrals we evaluate later in the course will require us to deal with integrals along certain curves in exactly this way. Be very careful about estimating the modulus of the denominator of the integrand-perhaps it is time to call on an old friend.
Transcribed Image Text:Question 4. Let R > 0 be a real number and let C denote the semicircular arc of radius R in the upper half of the complex plane centred at the origin from R to - R. - R Use lemma 3.2.9 to prove that Joh 1 2² +4 CH R R dz→0 as R→∞. It is vital to master the technique of evaluating the limiting value of integrals such as this many contour integrals we evaluate later in the course will require us to deal with integrals along certain curves in exactly this way. Be very careful about estimating the modulus of the denominator of the integrand-perhaps it is time to call on an old friend.
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