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Example Questions
例子问题# 1:Residue Theory
Cauchy's Residue Theorem is as follows:
Letbe a simple closed contour, described positively. If a functionis analytic insideexcept for a finite number of singular pointsinside, then
Brown, J. W., & Churchill, R. V. (2009).Complex variables and applications. Boston, MA: McGraw-Hill Higher Education.
Use Cauchy's Residue Theorem to evaluate the integral of
in the region.
Note, for
a singularity exists where. Thus, since whereis the only singularity forinside, we seek to evaluate the residue for.
观察,
The coefficient ofis.
Thus,
.
Therefore, by Cauchy's Residue Theorem,
Hence,
例子问题# 1:Residue Theory
Cauchy's Residue Theorem is as follows:
Letbe a simple closed contour, described positively. If a functionis analytic insideexcept for a finite number of singular pointsinside, then
Brown, J. W., & Churchill, R. V. (2009).Complex variables and applications. Boston, MA: McGraw-Hill Higher Education.
Using Cauchy's Residue Theorem, evaluate the integral of
in the region
Note, for
a singularity exists where. Thus, since whereis the only singularity forinside, we seek to evaluate the residue for.
观察,
The coefficient ofis.
Thus,
.
Therefore, by Cauchy's Residue Theorem,
Hence,
Example Question #42 :Complex Analysis
Cauchy's Residue Theorem is as follows:
Letbe a simple closed contour, described positively. If a functionis analytic insideexcept for a finite number of singular pointsinside, then
Brown, J. W., & Churchill, R. V. (2009).Complex variables and applications. Boston, MA: McGraw-Hill Higher Education.
Use Cauchy's Residue Theorem to evaluate the integral of
in the region.
Note, there is one singularity forwhere.
Let
Then
so
.
Therefore, there is one singularity forwhere. Hence, we seek to compute the residue forwhere
观察,
So, when,.
Thus, the coefficient ofis.
Therefore,
Hence, by Cauchy's Residue Theorem,
Therefore,
例子问题# 1:Residue Theory
Cauchy's Residue Theorem is as follows:
Letbe a simple closed contour, described positively. If a functionis analytic insideexcept for a finite number of singular pointsinside, then
For the following problem, use a modified version of the theorem which goes as follows:
Residue Theorem
If a functionis analytic everywhere in the finite plane except for a finite number of singular points interior to a positively oriented simple closed contour, then
Brown, J. W., & Churchill, R. V. (2009).Complex variables and applications. Boston, MA: McGraw-Hill Higher Education.
Use the Residue Theorem to evaluate the integral of
in the region.
0
0
Note,
Thus, seeking to apply the Residue Theorem above forinside, we evaluate the residue for.
观察,
The coefficient ofis.
Thus,
.
Therefore, by the Residue Theorem above,
Hence,
例子问题# 1:Residue Theory
Cauchy's Residue Theorem is as follows:
Letbe a simple closed contour, described positively. If a functionis analytic insideexcept for a finite number of singular pointsinside, then
For the following problem, use a modified version of the theorem which goes as follows:
Residue Theorem
If a functionis analytic everywhere in the finite plane except for a finite number of singular points interior to a positively oriented simple closed contour, then
Brown, J. W., & Churchill, R. V. (2009).Complex variables and applications. Boston, MA: McGraw-Hill Higher Education.
Use the Residue Theorem to evaluate the integral of
in the region.
Note,
Thus, seeking to apply the Residue Theorem above forinside, we evaluate the residue for.
观察,
The coefficient ofis.
Thus,
.
Therefore, by the Residue Theorem above,
Hence,
例子问题# 1:Residue Theory
Cauchy's Residue Theorem is as follows:
Letbe a simple closed contour, described positively. If a functionis analytic insideexcept for a finite number of singular pointsinside, then
For the following problem, use a modified version of the theorem which goes as follows:
Residue Theorem
If a functionis analytic everywhere in the finite plane except for a finite number of singular points interior to a positively oriented simple closed contour, then
Brown, J. W., & Churchill, R. V. (2009).Complex variables and applications. Boston, MA: McGraw-Hill Higher Education.
Use the Residue Theorem to evaluate the integral of
in the region.
Note,
Thus, seeking to apply the Residue Theorem above forinside, we evaluate the residue for.
观察,
The coefficient ofis.
Thus,
.
Therefore, by the Residue Theorem above,
Hence,
例子问题# 1:Residue Theory
Cauchy's Residue Theorem is as follows:
Letbe a simple closed contour, described positively. If a functionis analytic insideexcept for a finite number of singular pointsinside, then
For the following problem, use a modified version of the theorem which goes as follows:
Residue Theorem
If a functionis analytic everywhere in the finite plane except for a finite number of singular points interior to a positively oriented simple closed contour, then
Brown, J. W., & Churchill, R. V. (2009).Complex variables and applications. Boston, MA: McGraw-Hill Higher Education.
Use the Residue Theorem to evaluate the integral of
in the region.
Note,
Thus, seeking to apply the Residue Theorem above forinside, we evaluate the residue for.
观察, the coefficient ofis.
Thus,
.
Therefore, by the Residue Theorem above,
Hence,
例子问题# 1:Residue Theory
Cauchy's Residue Theorem is as follows:
Letbe a simple closed contour, described positively. If a functionis analytic insideexcept for a finite number of singular pointsinside, then
For the following problem, use a modified version of the theorem which goes as follows:
Residue Theorem
If a functionis analytic everywhere in the finite plane except for a finite number of singular points interior to a positively oriented simple closed contour, then
Brown, J. W., & Churchill, R. V. (2009).Complex variables and applications. Boston, MA: McGraw-Hill Higher Education.
Use the Residue Theorem to evaluate the integral of
in the region.
Note,
Thus, seeking to apply the Residue Theorem above forinside, we evaluate the residue for.
观察,
The coefficient ofis.
Thus,
.
Therefore, by the Residue Theorem above,
Hence,
例子问题# 1:Residue Theory
Find the residue of the function
.
观察
The coefficient ofis.
Thus,
.
Example Question #2 :Residue Theory
Find the residue atof
.
Let.
观察,
The coefficient ofissince there is noterm in the sum.
Thus,
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