Please solve the questions with complex analysis. These questions are integral questions in complex analysis. Please pay attention to the hints when solving the questions. Solve e) using the method indicated. Answer only d) and e).

Linear Algebra: A Modern Introduction
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ISBN:9781285463247
Author:David Poole
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Chapter6: Vector Spaces
Section6.6: The Matrix Of A Linear Transformation
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Please solve the questions with complex analysis. These questions are integral questions in complex analysis. Please pay attention to the hints when solving the questions. Solve e) using the method indicated. Answer only d) and e).

1- Assume that x, y e R and z = x + iy E C.Calculate the following integrals using the given contour
and assuming that R → 00, ɛ → 0 and Aø → 0.
∞ In(x)
dx
a- S.
1+x2
CR
b- dx
1+x2
C2
-R
R
C-
0.
dx
1+x2
o In(x)
(4+x²)2
co In² (x)
dx
2+x2
е-
(Hint 1: Use same strategy with last homework, which is 2ni Ɛ Residue = Sctetcetc.(...). Prove
that fa(...) = 0 as R → ∞. For S,(...) use z = ei® and evaluate the contribution from this term. For
dx then expand log (z2) function as (loglz| + iArg(z))*.)
In(x)
part e), first solve
(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 E C.Calculate the following integrals using the given contour and assuming that R → 00, ɛ → 0 and Aø → 0. ∞ In(x) dx a- S. 1+x2 CR b- dx 1+x2 C2 -R R C- 0. dx 1+x2 o In(x) (4+x²)2 co In² (x) dx 2+x2 е- (Hint 1: Use same strategy with last homework, which is 2ni Ɛ Residue = Sctetcetc.(...). Prove that fa(...) = 0 as R → ∞. For S,(...) use z = ei® and evaluate the contribution from this term. For dx then expand log (z2) function as (loglz| + iArg(z))*.) In(x) part e), first solve (Hint 2: Here for log and sqrt functions the principal branch definitions can be employed.)
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