Complex Analysis : Residue Theory

Study concepts, example questions & explanations for Complex Analysis

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Example Questions

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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 .

Possible Answers:

Correct answer:

Explanation:

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 

Possible Answers:

Correct answer:

Explanation:

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 .

Possible Answers:

Correct answer:

Explanation:

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 .

Possible Answers:

0

Correct answer:

0

Explanation:

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 .

Possible Answers:

Correct answer:

Explanation:

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 .

Possible Answers:

Correct answer:

Explanation:

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 .

Possible Answers:

Correct answer:

Explanation:

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 .

Possible Answers:

Correct answer:

Explanation:

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 

.

Possible Answers:

Correct answer:

Explanation:

Observe

The coefficient of  is .

Thus,

.

Example Question #10 : Residue Theory

Find the residue at  of 

.

Possible Answers:

Correct answer:

Explanation:

Let 

Observe,

The coefficient of  is  since there is no  term in the sum.

Thus, 

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