All Complex Analysis Resources
Example Questions
Example Question #1 : Residue Theory
Cauchy's Residue Theorem is as follows:
Let be a simple closed contour, described positively. If a function is analytic inside except for a finite number of singular points inside , 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 where is the only singularity for inside , we seek to evaluate the residue for .
Observe,
The coefficient of is .
Thus,
.
Therefore, by Cauchy's Residue Theorem,
Hence,
Example Question #2 : Residue Theory
Cauchy's Residue Theorem is as follows:
Let be a simple closed contour, described positively. If a function is analytic inside except for a finite number of singular points inside , 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 where is the only singularity for inside , we seek to evaluate the residue for .
Observe,
The coefficient of is .
Thus,
.
Therefore, by Cauchy's Residue Theorem,
Hence,
Example Question #3 : Residue Theory
Cauchy's Residue Theorem is as follows:
Let be a simple closed contour, described positively. If a function is analytic inside except for a finite number of singular points inside , 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 for where .
Let
Then
so
.
Therefore, there is one singularity for where . Hence, we seek to compute the residue for where
Observe,
So, when , .
Thus, the coefficient of is .
Therefore,
Hence, by Cauchy's Residue Theorem,
Therefore,
Example Question #4 : Residue Theory
Cauchy's Residue Theorem is as follows:
Let be a simple closed contour, described positively. If a function is analytic inside except for a finite number of singular points inside , then
For the following problem, use a modified version of the theorem which goes as follows:
Residue Theorem
If a function is 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 for inside , we evaluate the residue for .
Observe,
The coefficient of is .
Thus,
.
Therefore, by the Residue Theorem above,
Hence,
Example Question #5 : Residue Theory
Cauchy's Residue Theorem is as follows:
Let be a simple closed contour, described positively. If a function is analytic inside except for a finite number of singular points inside , then
For the following problem, use a modified version of the theorem which goes as follows:
Residue Theorem
If a function is 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 for inside , we evaluate the residue for .
Observe,
The coefficient of is .
Thus,
.
Therefore, by the Residue Theorem above,
Hence,
Example Question #6 : Residue Theory
Cauchy's Residue Theorem is as follows:
Let be a simple closed contour, described positively. If a function is analytic inside except for a finite number of singular points inside , then
For the following problem, use a modified version of the theorem which goes as follows:
Residue Theorem
If a function is 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 for inside , we evaluate the residue for .
Observe,
The coefficient of is .
Thus,
.
Therefore, by the Residue Theorem above,
Hence,
Example Question #7 : Residue Theory
Cauchy's Residue Theorem is as follows:
Let be a simple closed contour, described positively. If a function is analytic inside except for a finite number of singular points inside , then
For the following problem, use a modified version of the theorem which goes as follows:
Residue Theorem
If a function is 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 for inside , we evaluate the residue for .
Observe, the coefficient of is .
Thus,
.
Therefore, by the Residue Theorem above,
Hence,
Example Question #8 : Residue Theory
Cauchy's Residue Theorem is as follows:
Let be a simple closed contour, described positively. If a function is analytic inside except for a finite number of singular points inside , then
For the following problem, use a modified version of the theorem which goes as follows:
Residue Theorem
If a function is 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 for inside , we evaluate the residue for .
Observe,
The coefficient of is .
Thus,
.
Therefore, by the Residue Theorem above,
Hence,
Example Question #9 : Residue Theory
Find the residue of the function
.
Observe
The coefficient of is .
Thus,
.
Example Question #1 : Residue Theory
Find the residue at of
.
Let .
Observe,
The coefficient of is since there is no term in the sum.
Thus,