For each statement, answer either true or false and include a brief/simple justification. We are NOT allowed to use the Residue Theorem here. We are limited to using Cauchy's Formula for this integral (see below). This is for a course in complex analysis.
For each statement, answer either true or false and include a brief/simple justification. We are NOT allowed to use the Residue Theorem here. We are limited to using Cauchy's Formula for this integral (see below). This is for a course in complex analysis.
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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![For each statement, answer either true or
false and include a brief/simple
justification. We are NOT allowed to use
the Residue Theorem here. We are
limited to using Cauchy's Formula for this
integral (see below). This is for a course in
complex analysis.
(10) S₁²=1 e ² dz = 2πi.
√₁2₁=1
Cauchy's Formula
THEOREM 4 Cauchy's Formula Suppose that f is analytic on a domain D and that y is a
piecewise smooth, positively oriented simple closed curve in D whose inside also
lies in D. Then
f(z) =
1 [ f()
5-2
Σπί
g() =
2.3 Cauchy's Theorem and Cauchy's Formula 111
Proof Since z is inside 2, there is a positive number &, so small that the disc
{w: |wz|< do} lies within also. Let & be a positive number smaller than do,
and let 2, be the domain obtained by deleting the disc {w: w-zl<8} from 2.
The boundary of , consists of the two curves y, oriented positively, and the
circle {w: |wz| = 8}, oriented negatively. We may apply Green's Theorem to the
function
dt, for all z € Ω.
dg
ax
on the region 2. Because g is analytic on ,, it follows (as in the proof of Theorem
1) that
+1
f(5)
5-2
dg
ду
throughout ,. Hence, the double integral term is zero, so
=
0-₂ [+]dx dy-fº
0=
=0
g(t) dt](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffc7689b9-c4fe-4bee-b670-05d15f3915cb%2F354a02e6-dcdb-42dd-9735-35daa4bd3869%2Fx5wg1v.jpeg&w=3840&q=75)
Transcribed Image Text:For each statement, answer either true or
false and include a brief/simple
justification. We are NOT allowed to use
the Residue Theorem here. We are
limited to using Cauchy's Formula for this
integral (see below). This is for a course in
complex analysis.
(10) S₁²=1 e ² dz = 2πi.
√₁2₁=1
Cauchy's Formula
THEOREM 4 Cauchy's Formula Suppose that f is analytic on a domain D and that y is a
piecewise smooth, positively oriented simple closed curve in D whose inside also
lies in D. Then
f(z) =
1 [ f()
5-2
Σπί
g() =
2.3 Cauchy's Theorem and Cauchy's Formula 111
Proof Since z is inside 2, there is a positive number &, so small that the disc
{w: |wz|< do} lies within also. Let & be a positive number smaller than do,
and let 2, be the domain obtained by deleting the disc {w: w-zl<8} from 2.
The boundary of , consists of the two curves y, oriented positively, and the
circle {w: |wz| = 8}, oriented negatively. We may apply Green's Theorem to the
function
dt, for all z € Ω.
dg
ax
on the region 2. Because g is analytic on ,, it follows (as in the proof of Theorem
1) that
+1
f(5)
5-2
dg
ду
throughout ,. Hence, the double integral term is zero, so
=
0-₂ [+]dx dy-fº
0=
=0
g(t) dt
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