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1- Assume that x, y € R and z = x + iy € c.Calculate the following integrals using the given contour and assuming that R → 00, e → 0 and Ap → 0. 0o In(x) dx 1+x2 а- CR b- S. dx Jo 1+x2 -R 00 Vx

1- Assume that x, y € R and z = x + iy € c.Calculate the following integrals using the given contour and assuming that R → 00, e → 0 and Ap → 0. 0o In(x) dx 1+x2 а- CR b- S. dx Jo 1+x2 -R 00 Vx

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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1- Assume that x, y E R and z = x+ iy € C.Calculate the following integrals using the given contour and assuming that R - 00, 8 - 0 and Aø - 0. 0o In(x) Jo Itx2 a- CR 0 Vx b- So CE 0 1+x2 C2 -R -E 00 Vx c- So 1+x2 0o In(x) d- So (4+x?)2 co In² (x) е- 2+x2 2ni E Residue = Sc +e,+og+G,(). Prove co. For S,(.) use z = egi® and evaluate the contribution from this term. For (Hint 1: Use that Sa(..) = 0 as R → part e), first solve dx then expand log (2) function as (log|z| + iArg(=)*) In(x) 2+x2 (Hint 2: Here for log and sqrt functions the principal branch definitions can be employed.)
Transcribed Image Text:1- Assume that x, y E R and z = x+ iy € C.Calculate the following integrals using the given contour and assuming that R - 00, 8 - 0 and Aø - 0. 0o In(x) Jo Itx2 a- CR 0 Vx b- So CE 0 1+x2 C2 -R -E 00 Vx c- So 1+x2 0o In(x) d- So (4+x?)2 co In² (x) е- 2+x2 2ni E Residue = Sc +e,+og+G,(). Prove co. For S,(.) use z = egi® and evaluate the contribution from this term. For (Hint 1: Use that Sa(..) = 0 as R → part e), first solve dx then expand log (2) function as (log|z| + iArg(=)*) In(x) 2+x2 (Hint 2: Here for log and sqrt functions the principal branch definitions can be employed.)
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