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In(x) d- 'o (4+x²)² Lo In?x) o 2+x2 00 е- dx

In(x) d- 'o (4+x²)² Lo In?x) o 2+x2 00 е- dx

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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These are Complex Analysis Questions. Please answer by HINT 1 and  HINT 2. Please solve only d) and e). For e), there is a method in HINT1. 

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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