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Assume that x,y € R and z=x+iy € C. Calculate the following integrals using the given contour and assuming that R→∞, € →0 and Δ?→0. (SOLVE THE LAST TWO SUBPARTS.)

Assume that x,y € R and z=x+iy € C. Calculate the following integrals using the given contour and assuming that R→∞, € →0 and Δ?→0. (SOLVE THE LAST TWO SUBPARTS.)

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter6: Vector Spaces
Section6.6: The Matrix Of A Linear Transformation
Problem 30EQ
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Assume that x,y € R and z=x+iy € C. Calculate the following integrals using the given contour and assuming that R→∞, € →0 and Δ?→0. (SOLVE THE LAST TWO SUBPARTS.)

Assume that x, y €R and z = x + iy € C .Calculate the following integrals using the given contour and assuming that R → 0o, e – 0 and Ao 0. (Hint 1: Use 2ri E Residue = Sc,+c,+Cg+c, ...). Prove that Sa(...) 0 as R → co. For S,(. use z = et0 and evaluate the contribution from this term. For part e), first solve dx then expand log?(z) function rco In(x) as (log|z| + iArg(z))°.) (Hint 2: Here for log and sqrt functions the principal branch definitions can be employed.) CR C2 -R -E In(x) ro∞ In(x) а- d- Jo (4+x²)² xp Z 1+x2 o Vx b- Jo T+x² dx S° In?cx) e- Jo 2+x² dx Jo I+x? C- 1+x2
Transcribed Image Text:Assume that x, y €R and z = x + iy € C .Calculate the following integrals using the given contour and assuming that R → 0o, e – 0 and Ao 0. (Hint 1: Use 2ri E Residue = Sc,+c,+Cg+c, ...). Prove that Sa(...) 0 as R → co. For S,(. use z = et0 and evaluate the contribution from this term. For part e), first solve dx then expand log?(z) function rco In(x) as (log|z| + iArg(z))°.) (Hint 2: Here for log and sqrt functions the principal branch definitions can be employed.) CR C2 -R -E In(x) ro∞ In(x) а- d- Jo (4+x²)² xp Z 1+x2 o Vx b- Jo T+x² dx S° In?cx) e- Jo 2+x² dx Jo I+x? C- 1+x2
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