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Define f(z): {ze C: Imz>0} → Cas √zeiz f(z) = 2²+1' where √ above denotes the branch of the (double-valued) function log which is continuous in the upper half-plane. (i) Calculate p.v. f(x) dr. (ii) Prove that S VT(cosx+sinx) x²+1 П dx = √2e Remark 1. Note that f(z) has a singularity at 0. Moreover, 0 is not an isolated singularity of f(2). But you can still overcome this problem by introducing a small semicircle around 0. Make sure that you justify your limits. Remark 2. The branch √ above can be written explicitly as {' 2 Log if Im z≥0, 20 0 if z=0, where = Logz In|z|+i Arg z with - < Arg z < く Зп

Define f(z): {ze C: Imz>0} → Cas √zeiz f(z) = 2²+1' where √ above denotes the branch of the (double-valued) function log which is continuous in the upper half-plane. (i) Calculate p.v. f(x) dr. (ii) Prove that S VT(cosx+sinx) x²+1 П dx = √2e Remark 1. Note that f(z) has a singularity at 0. Moreover, 0 is not an isolated singularity of f(2). But you can still overcome this problem by introducing a small semicircle around 0. Make sure that you justify your limits. Remark 2. The branch √ above can be written explicitly as {' 2 Log if Im z≥0, 20 0 if z=0, where = Logz In|z|+i Arg z with - < Arg z < く Зп

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.5: Rational Functions
Problem 50E
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Define f(z): {ze C: Imz>0} → Cas √zeiz f(z) = 2²+1' where √ above denotes the branch of the (double-valued) function log which is continuous in the upper half-plane. (i) Calculate p.v. f(x) dr. (ii) Prove that S VT(cosx+sinx) x²+1 П dx = √2e Remark 1. Note that f(z) has a singularity at 0. Moreover, 0 is not an isolated singularity of f(2). But you can still overcome this problem by introducing a small semicircle around 0. Make sure that you justify your limits. Remark 2. The branch √ above can be written explicitly as {' 2 Log if Im z≥0, 20 0 if z=0, where = Logz In|z|+i Arg z with - < Arg z < く Зп
Transcribed Image Text:Define f(z): {ze C: Imz>0} → Cas √zeiz f(z) = 2²+1' where √ above denotes the branch of the (double-valued) function log which is continuous in the upper half-plane. (i) Calculate p.v. f(x) dr. (ii) Prove that S VT(cosx+sinx) x²+1 П dx = √2e Remark 1. Note that f(z) has a singularity at 0. Moreover, 0 is not an isolated singularity of f(2). But you can still overcome this problem by introducing a small semicircle around 0. Make sure that you justify your limits. Remark 2. The branch √ above can be written explicitly as {' 2 Log if Im z≥0, 20 0 if z=0, where = Logz In|z|+i Arg z with - < Arg z < く Зп
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