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 #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.
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 #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, 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 #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
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 #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
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 #51 : Complex Analysis
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 #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
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 #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
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 #51 : Complex Analysis
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,
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