Calculate the integral of exp(z)/sin(z) over |z|=4 using the residue theorem.
Click to view the integral in correct format.
Calculate the integral of exp(z)/sin(z) (as in the image above) over the positively oriented circle defined by |z|=4 using the residue theorem.
This is a question that I'm just not sure I'm doing completely correct and would like some confirmation or correction. Pictures of my work are added below.
From exp(z)/sin(z), the singularities within |z|=4 are 0, pi, and -pi where pi is in reference to the 3.14... pi. I used the residue theorem with the p(z) and q(z) standards where Res (f, z0) = p(z0)/q'(zo) to get
Res (f, 0) = exp(z)/cos(z) | (z=0) = 1
Res (f, pi) = exp(z)/cos(z) | (z=pi) = -e^(pi)
Res (f, -pi) = exp(z)/cos(z) | (z=-pi) = -e^(-pi)
Thus, the residue theorem yields the (unsimplified) answer of 2(pi)(i)(1-e^(pi)-e^(-pi)).
Click to view the guidelines I assumed to start my calculation.
Click to view my calculations and answer in a more legible format.
complex-analysis exponential-function mathematical-physics trigonometric-integrals several-complex-variables
asked Nov 27 at 10:48
Josie Evans
162
add a comment |
Click to view the integral in correct format.
Calculate the integral of exp(z)/sin(z) (as in the image above) over the positively oriented circle defined by |z|=4 using the residue theorem.
This is a question that I'm just not sure I'm doing completely correct and would like some confirmation or correction. Pictures of my work are added below.
From exp(z)/sin(z), the singularities within |z|=4 are 0, pi, and -pi where pi is in reference to the 3.14... pi. I used the residue theorem with the p(z) and q(z) standards where Res (f, z0) = p(z0)/q'(zo) to get
Res (f, 0) = exp(z)/cos(z) | (z=0) = 1
Res (f, pi) = exp(z)/cos(z) | (z=pi) = -e^(pi)
Res (f, -pi) = exp(z)/cos(z) | (z=-pi) = -e^(-pi)
Thus, the residue theorem yields the (unsimplified) answer of 2(pi)(i)(1-e^(pi)-e^(-pi)).
Click to view the guidelines I assumed to start my calculation.
Click to view my calculations and answer in a more legible format.
complex-analysis exponential-function mathematical-physics trigonometric-integrals several-complex-variables
asked Nov 27 at 10:48
Josie Evans
162
Hi and welcome to the site! Since this is a site that encourages and helps with learning, it is best if you show your own ideas and efforts in solving the question. Can you edit your question to add your thoughts and ideas about it? Don't worry if it's wrong - that's what we're here for.
– 5xum
Nov 27 at 10:49
Also, don't get discouraged by the downvote. I downvoted the question and voted to close it because at the moment, it is not up to site standards (you have shown no work you did on your own). If you edit your question so that you show what you tried and how far you got, I will not only remove the downvote, I will add an upvote.
– 5xum
Nov 27 at 10:49
1
Thank you for getting me up to speed with the site standards! I've added my calculations as well as links to my work in a more readable fashion. Hope this satisfies the guidelines and helps to see where I might be going wrong in my work!
– Josie Evans
Nov 27 at 11:35
As promised, I retracted my close vote :)
– 5xum
Nov 27 at 11:40
add a comment |
Click to view the integral in correct format.
Calculate the integral of exp(z)/sin(z) (as in the image above) over the positively oriented circle defined by |z|=4 using the residue theorem.
This is a question that I'm just not sure I'm doing completely correct and would like some confirmation or correction. Pictures of my work are added below.
From exp(z)/sin(z), the singularities within |z|=4 are 0, pi, and -pi where pi is in reference to the 3.14... pi. I used the residue theorem with the p(z) and q(z) standards where Res (f, z0) = p(z0)/q'(zo) to get
Res (f, 0) = exp(z)/cos(z) | (z=0) = 1
Res (f, pi) = exp(z)/cos(z) | (z=pi) = -e^(pi)
Res (f, -pi) = exp(z)/cos(z) | (z=-pi) = -e^(-pi)
Thus, the residue theorem yields the (unsimplified) answer of 2(pi)(i)(1-e^(pi)-e^(-pi)).
Click to view the guidelines I assumed to start my calculation.
Click to view my calculations and answer in a more legible format.
complex-analysis exponential-function mathematical-physics trigonometric-integrals several-complex-variables
asked Nov 27 at 10:48
Josie Evans
162
Click to view the integral in correct format.
Calculate the integral of exp(z)/sin(z) (as in the image above) over the positively oriented circle defined by |z|=4 using the residue theorem.
This is a question that I'm just not sure I'm doing completely correct and would like some confirmation or correction. Pictures of my work are added below.
From exp(z)/sin(z), the singularities within |z|=4 are 0, pi, and -pi where pi is in reference to the 3.14... pi. I used the residue theorem with the p(z) and q(z) standards where Res (f, z0) = p(z0)/q'(zo) to get
Res (f, 0) = exp(z)/cos(z) | (z=0) = 1
Res (f, pi) = exp(z)/cos(z) | (z=pi) = -e^(pi)
Res (f, -pi) = exp(z)/cos(z) | (z=-pi) = -e^(-pi)
Thus, the residue theorem yields the (unsimplified) answer of 2(pi)(i)(1-e^(pi)-e^(-pi)).
Click to view the guidelines I assumed to start my calculation.
Click to view my calculations and answer in a more legible format.
complex-analysis exponential-function mathematical-physics trigonometric-integrals several-complex-variables
complex-analysis exponential-function mathematical-physics trigonometric-integrals several-complex-variables
asked Nov 27 at 10:48
Josie Evans
162
asked Nov 27 at 10:48
Josie Evans
162
edited Nov 27 at 11:31
asked Nov 27 at 10:48
Josie Evans
162
asked Nov 27 at 10:48
Josie Evans
162
asked Nov 27 at 10:48
Josie Evans
162
162
Hi and welcome to the site! Since this is a site that encourages and helps with learning, it is best if you show your own ideas and efforts in solving the question. Can you edit your question to add your thoughts and ideas about it? Don't worry if it's wrong - that's what we're here for.
– 5xum
Nov 27 at 10:49
Also, don't get discouraged by the downvote. I downvoted the question and voted to close it because at the moment, it is not up to site standards (you have shown no work you did on your own). If you edit your question so that you show what you tried and how far you got, I will not only remove the downvote, I will add an upvote.
– 5xum
Nov 27 at 10:49
1
Thank you for getting me up to speed with the site standards! I've added my calculations as well as links to my work in a more readable fashion. Hope this satisfies the guidelines and helps to see where I might be going wrong in my work!
– Josie Evans
Nov 27 at 11:35
As promised, I retracted my close vote :)
– 5xum
Nov 27 at 11:40
add a comment |
Hi and welcome to the site! Since this is a site that encourages and helps with learning, it is best if you show your own ideas and efforts in solving the question. Can you edit your question to add your thoughts and ideas about it? Don't worry if it's wrong - that's what we're here for.
– 5xum
Nov 27 at 10:49
Also, don't get discouraged by the downvote. I downvoted the question and voted to close it because at the moment, it is not up to site standards (you have shown no work you did on your own). If you edit your question so that you show what you tried and how far you got, I will not only remove the downvote, I will add an upvote.
– 5xum
Nov 27 at 10:49
1
Thank you for getting me up to speed with the site standards! I've added my calculations as well as links to my work in a more readable fashion. Hope this satisfies the guidelines and helps to see where I might be going wrong in my work!
– Josie Evans
Nov 27 at 11:35
As promised, I retracted my close vote :)
– 5xum
Nov 27 at 11:40
Hi and welcome to the site! Since this is a site that encourages and helps with learning, it is best if you show your own ideas and efforts in solving the question. Can you edit your question to add your thoughts and ideas about it? Don't worry if it's wrong - that's what we're here for.
– 5xum
Nov 27 at 10:49
Hi and welcome to the site! Since this is a site that encourages and helps with learning, it is best if you show your own ideas and efforts in solving the question. Can you edit your question to add your thoughts and ideas about it? Don't worry if it's wrong - that's what we're here for.
– 5xum
Nov 27 at 10:49
Also, don't get discouraged by the downvote. I downvoted the question and voted to close it because at the moment, it is not up to site standards (you have shown no work you did on your own). If you edit your question so that you show what you tried and how far you got, I will not only remove the downvote, I will add an upvote.
– 5xum
Nov 27 at 10:49
Also, don't get discouraged by the downvote. I downvoted the question and voted to close it because at the moment, it is not up to site standards (you have shown no work you did on your own). If you edit your question so that you show what you tried and how far you got, I will not only remove the downvote, I will add an upvote.
– 5xum
Nov 27 at 10:49
1
1
Thank you for getting me up to speed with the site standards! I've added my calculations as well as links to my work in a more readable fashion. Hope this satisfies the guidelines and helps to see where I might be going wrong in my work!
– Josie Evans
Nov 27 at 11:35
Thank you for getting me up to speed with the site standards! I've added my calculations as well as links to my work in a more readable fashion. Hope this satisfies the guidelines and helps to see where I might be going wrong in my work!
– Josie Evans
Nov 27 at 11:35
As promised, I retracted my close vote :)
– 5xum
Nov 27 at 11:40
As promised, I retracted my close vote :)
– 5xum
Nov 27 at 11:40
add a comment |
1 Answer
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Your computations of the residues are correct. And the computation of the integral (via residue theorem) also looks fine.
answered Nov 27 at 12:02
Fred
44.2k1845
add a comment |
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1 Answer
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Your computations of the residues are correct. And the computation of the integral (via residue theorem) also looks fine.
answered Nov 27 at 12:02
Fred
44.2k1845
add a comment |
Your computations of the residues are correct. And the computation of the integral (via residue theorem) also looks fine.
answered Nov 27 at 12:02
Fred
44.2k1845
add a comment |
Your computations of the residues are correct. And the computation of the integral (via residue theorem) also looks fine.
answered Nov 27 at 12:02
Fred
44.2k1845
Your computations of the residues are correct. And the computation of the integral (via residue theorem) also looks fine.
answered Nov 27 at 12:02
Fred
44.2k1845
answered Nov 27 at 12:02
Fred
44.2k1845
answered Nov 27 at 12:02
Fred
44.2k1845
answered Nov 27 at 12:02
Fred
44.2k1845
44.2k1845
add a comment |
add a comment |
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Hi and welcome to the site! Since this is a site that encourages and helps with learning, it is best if you show your own ideas and efforts in solving the question. Can you edit your question to add your thoughts and ideas about it? Don't worry if it's wrong - that's what we're here for.
– 5xum
Nov 27 at 10:49
Also, don't get discouraged by the downvote. I downvoted the question and voted to close it because at the moment, it is not up to site standards (you have shown no work you did on your own). If you edit your question so that you show what you tried and how far you got, I will not only remove the downvote, I will add an upvote.
– 5xum
Nov 27 at 10:49
1
Thank you for getting me up to speed with the site standards! I've added my calculations as well as links to my work in a more readable fashion. Hope this satisfies the guidelines and helps to see where I might be going wrong in my work!
– Josie Evans
Nov 27 at 11:35
As promised, I retracted my close vote :)
– 5xum
Nov 27 at 11:40