Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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Show that the function f(x) = sin(x)/x has a removable singularity. What are the left and right handed limits?
18.9. Let denote the boundary of the rectangle whose vertices are
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(a).
之一
dz, (b).
dz, (b).
COS 2
coz dz,
dz
(z+1)
(d).
z 2 +2
dz, (e).
(c). (2z+1)zdz,
z+
1
(f). £,
· [e² sin = + (2² + 3)²] dz.
(2+3)2
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0<|z|< 1,
f(E)
f(E)
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--- d.
Chapter 16 Solutions
Advanced Engineering Mathematics
Ch. 16.1 - Prob. 1PCh. 16.1 - Prob. 2PCh. 16.1 - Prob. 3PCh. 16.1 - Prob. 4PCh. 16.1 - Prob. 5PCh. 16.1 - Prob. 6PCh. 16.1 - Prob. 7PCh. 16.1 - Prob. 8PCh. 16.1 - Prob. 9PCh. 16.1 - Prob. 10P
Ch. 16.1 - Prob. 11PCh. 16.1 - Prob. 12PCh. 16.1 - Prob. 13PCh. 16.1 - Prob. 14PCh. 16.1 - Prob. 15PCh. 16.1 - Prob. 16PCh. 16.1 - Prob. 18PCh. 16.1 - Prob. 19PCh. 16.1 - Prob. 20PCh. 16.1 - Prob. 21PCh. 16.1 - Prob. 22PCh. 16.1 - Prob. 23PCh. 16.1 - Prob. 24PCh. 16.1 - Prob. 25PCh. 16.2 - Prob. 1PCh. 16.2 - Prob. 2PCh. 16.2 - Prob. 3PCh. 16.2 - Prob. 4PCh. 16.2 - Prob. 5PCh. 16.2 - Prob. 6PCh. 16.2 - Prob. 7PCh. 16.2 - Prob. 8PCh. 16.2 - Prob. 9PCh. 16.2 - Prob. 10PCh. 16.2 - Prob. 11PCh. 16.2 - Prob. 12PCh. 16.2 - Prob. 13PCh. 16.2 - Prob. 14PCh. 16.2 - Prob. 15PCh. 16.2 - Prob. 16PCh. 16.2 - Prob. 17PCh. 16.2 - Prob. 18PCh. 16.2 - Prob. 19PCh. 16.2 - Prob. 20PCh. 16.2 - Prob. 21PCh. 16.2 - Prob. 22PCh. 16.2 - Prob. 23PCh. 16.2 - Prob. 24PCh. 16.3 - Prob. 1PCh. 16.3 - Prob. 2PCh. 16.3 - Prob. 3PCh. 16.3 - Prob. 4PCh. 16.3 - Prob. 5PCh. 16.3 - Prob. 6PCh. 16.3 - Prob. 7PCh. 16.3 - Prob. 8PCh. 16.3 - Prob. 9PCh. 16.3 - Prob. 10PCh. 16.3 - Prob. 11PCh. 16.3 - Prob. 12PCh. 16.3 - Prob. 14PCh. 16.3 - Prob. 15PCh. 16.3 - Prob. 16PCh. 16.3 - Prob. 17PCh. 16.3 - Prob. 18PCh. 16.3 - Prob. 19PCh. 16.3 - Prob. 20PCh. 16.3 - Prob. 21PCh. 16.3 - Prob. 22PCh. 16.3 - Prob. 23PCh. 16.3 - Prob. 24PCh. 16.3 - Prob. 25PCh. 16.4 - Prob. 1PCh. 16.4 - Prob. 2PCh. 16.4 - Prob. 3PCh. 16.4 - Prob. 4PCh. 16.4 - Prob. 5PCh. 16.4 - Prob. 6PCh. 16.4 - Prob. 7PCh. 16.4 - Prob. 8PCh. 16.4 - Prob. 9PCh. 16.4 - Prob. 10PCh. 16.4 - Prob. 11PCh. 16.4 - Prob. 12PCh. 16.4 - Prob. 13PCh. 16.4 - Prob. 14PCh. 16.4 - Prob. 15PCh. 16.4 - Prob. 16PCh. 16.4 - Prob. 17PCh. 16.4 - Prob. 18PCh. 16.4 - Prob. 19PCh. 16.4 - Prob. 20PCh. 16.4 - Prob. 21PCh. 16.4 - Prob. 22PCh. 16.4 - Prob. 23PCh. 16.4 - Prob. 24PCh. 16.4 - Prob. 25PCh. 16.4 - Prob. 26PCh. 16.4 - Prob. 28PCh. 16 - Prob. 1RQCh. 16 - Prob. 2RQCh. 16 - Prob. 3RQCh. 16 - Prob. 4RQCh. 16 - Prob. 5RQCh. 16 - Prob. 6RQCh. 16 - Prob. 7RQCh. 16 - Prob. 8RQCh. 16 - Prob. 9RQCh. 16 - Prob. 10RQCh. 16 - Prob. 11RQCh. 16 - Prob. 12RQCh. 16 - Prob. 13RQCh. 16 - Prob. 14RQCh. 16 - Prob. 15RQCh. 16 - Prob. 16RQCh. 16 - Prob. 17RQCh. 16 - Prob. 18RQCh. 16 - Prob. 19RQCh. 16 - Prob. 20RQCh. 16 - Prob. 21RQCh. 16 - Prob. 22RQCh. 16 - Prob. 23RQCh. 16 - Prob. 24RQCh. 16 - Prob. 25RQ
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- 18.4. Let f be analytic within and on a positively oriented closed contoury, and the point zo is not on y. Show that L f(z) (-20)2 dz = '(2) dz. 2-20arrow_forward18.9. Let denote the boundary of the rectangle whose vertices are -2-2i, 2-21,2+i and -2+i in the positive direction. Evaluate each of the following integrals: (a). rdz, (b). dz (b). COS 2 coz dz, (z+1) (d). 之一 z 2 +2 dz, (e). dz (c). (2z + 1)2dz, (2z+1) 1 (f). £, · [e² sin = + (2² + 3)²] dz. z (22+3)2arrow_forward18.8. (a). Let be the contour z = e-≤0≤ traversed in the า -dz = 2xi. positive direction. Show that, for any real constant a, Lex dzarrow_forward
- f(z) 18.7. Let f(z) = (e² + e³)/2. Evaluate dz, where y is any simple closed curve enclosing 0.arrow_forward18. If m n compute the gcd (a² + 1, a² + 1) in terms of a. [Hint: Let A„ = a² + 1 and show that A„|(Am - 2) if m > n.]arrow_forwardFor each real-valued nonprincipal character x mod k, let A(n) = x(d) and F(x) = Σ : dn * Prove that F(x) = L(1,x) log x + O(1). narrow_forwardBy considering appropriate series expansions, e². e²²/2. e²³/3. .... = = 1 + x + x² + · ... when |x| < 1. By expanding each individual exponential term on the left-hand side the coefficient of x- 19 has the form and multiplying out, 1/19!1/19+r/s, where 19 does not divide s. Deduce that 18! 1 (mod 19).arrow_forwardBy considering appropriate series expansions, ex · ex²/2 . ¸²³/³ . . .. = = 1 + x + x² +…… when |x| < 1. By expanding each individual exponential term on the left-hand side and multiplying out, show that the coefficient of x 19 has the form 1/19!+1/19+r/s, where 19 does not divide s.arrow_forwardLet 1 1 r 1+ + + 2 3 + = 823 823s Without calculating the left-hand side, prove that r = s (mod 823³).arrow_forwardFor each real-valued nonprincipal character X mod 16, verify that L(1,x) 0.arrow_forward*Construct a table of values for all the nonprincipal Dirichlet characters mod 16. Verify from your table that Σ x(3)=0 and Χ mod 16 Σ χ(11) = 0. x mod 16arrow_forwardFor each real-valued nonprincipal character x mod 16, verify that A(225) > 1. (Recall that A(n) = Σx(d).) d\narrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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