Calculating the residue of $frac{10z^4-10sin(z)}{z^3}, z(0) = 0$
up vote
3
down vote
favorite
$$frac{10z^4-10sin(z)}{z^3}, quad z(0) = 0.$$
I've gotten that $$operatorname{Res} = 0$$ but I'm not quite sure if that is correct, or if I have even used a correct pathway towards it. How should one work around sine (or cosine, for that matter) during residue calculations?
complex-analysis residue-calculus
edited Nov 21 at 12:25
Tianlalu
2,9801936
asked Oct 25 at 22:49
TootsieRoll
163
add a comment |
up vote
3
down vote
favorite
$$frac{10z^4-10sin(z)}{z^3}, quad z(0) = 0.$$
I've gotten that $$operatorname{Res} = 0$$ but I'm not quite sure if that is correct, or if I have even used a correct pathway towards it. How should one work around sine (or cosine, for that matter) during residue calculations?
complex-analysis residue-calculus
edited Nov 21 at 12:25
Tianlalu
2,9801936
asked Oct 25 at 22:49
TootsieRoll
163
Can you show us the pathway you chose so we can compare it with what we would do?
– Oscar Lanzi
Oct 26 at 0:15
add a comment |
up vote
3
down vote
favorite
up vote
3
down vote
favorite
$$frac{10z^4-10sin(z)}{z^3}, quad z(0) = 0.$$
I've gotten that $$operatorname{Res} = 0$$ but I'm not quite sure if that is correct, or if I have even used a correct pathway towards it. How should one work around sine (or cosine, for that matter) during residue calculations?
complex-analysis residue-calculus
edited Nov 21 at 12:25
Tianlalu
2,9801936
asked Oct 25 at 22:49
TootsieRoll
163
$$frac{10z^4-10sin(z)}{z^3}, quad z(0) = 0.$$
I've gotten that $$operatorname{Res} = 0$$ but I'm not quite sure if that is correct, or if I have even used a correct pathway towards it. How should one work around sine (or cosine, for that matter) during residue calculations?
complex-analysis residue-calculus
complex-analysis residue-calculus
edited Nov 21 at 12:25
Tianlalu
2,9801936
asked Oct 25 at 22:49
TootsieRoll
163
edited Nov 21 at 12:25
Tianlalu
2,9801936
asked Oct 25 at 22:49
TootsieRoll
163
edited Nov 21 at 12:25
Tianlalu
2,9801936
edited Nov 21 at 12:25
Tianlalu
2,9801936
edited Nov 21 at 12:25
Tianlalu
2,9801936
2,9801936
asked Oct 25 at 22:49
TootsieRoll
163
asked Oct 25 at 22:49
TootsieRoll
163
asked Oct 25 at 22:49
TootsieRoll
163
163
Can you show us the pathway you chose so we can compare it with what we would do?
– Oscar Lanzi
Oct 26 at 0:15
add a comment |
Can you show us the pathway you chose so we can compare it with what we would do?
– Oscar Lanzi
Oct 26 at 0:15
Can you show us the pathway you chose so we can compare it with what we would do?
– Oscar Lanzi
Oct 26 at 0:15
Can you show us the pathway you chose so we can compare it with what we would do?
– Oscar Lanzi
Oct 26 at 0:15
add a comment |
2 Answers
2
active
oldest
votes
up vote
1
down vote
Your answer is correct. The residue of a function is the coefficient of $z^{-1}$ of the Laurent series expansion of the function.
We obtain
begin{align*}
color{blue}{[z^{-1}]}&color{blue}{frac{10z^4-10sin z}{z^3}}\
&=[z^{-1}]left(10z-frac{10}{z^3}left(z-frac{z^3}{3!}+frac{z^5}{5!}-cdotsright)right)\
&=-10[z^{-1}]left(frac{1}{z^2}-frac{1}{3!}+frac{z^2}{5!}-cdotsright)\
&,,color{blue}{=0}
end{align*}
answered Oct 26 at 20:29
Markus Scheuer
59.5k455142
add a comment |
up vote
0
down vote
One way to work with sines and cosines in this problem is to reckon with their power series expansions found in many places, for instance here. You might also note that an even function has only even powers in its Laurent series whereas the residue comes from the $(-1)$ power, therefore even functions will have a residue of zero. Hmm, can you see why I would mention that last fact?
answered Oct 26 at 0:22
Oscar Lanzi
11.9k11936
add a comment |
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
Your answer is correct. The residue of a function is the coefficient of $z^{-1}$ of the Laurent series expansion of the function.
We obtain
begin{align*}
color{blue}{[z^{-1}]}&color{blue}{frac{10z^4-10sin z}{z^3}}\
&=[z^{-1}]left(10z-frac{10}{z^3}left(z-frac{z^3}{3!}+frac{z^5}{5!}-cdotsright)right)\
&=-10[z^{-1}]left(frac{1}{z^2}-frac{1}{3!}+frac{z^2}{5!}-cdotsright)\
&,,color{blue}{=0}
end{align*}
answered Oct 26 at 20:29
Markus Scheuer
59.5k455142
add a comment |
up vote
1
down vote
Your answer is correct. The residue of a function is the coefficient of $z^{-1}$ of the Laurent series expansion of the function.
We obtain
begin{align*}
color{blue}{[z^{-1}]}&color{blue}{frac{10z^4-10sin z}{z^3}}\
&=[z^{-1}]left(10z-frac{10}{z^3}left(z-frac{z^3}{3!}+frac{z^5}{5!}-cdotsright)right)\
&=-10[z^{-1}]left(frac{1}{z^2}-frac{1}{3!}+frac{z^2}{5!}-cdotsright)\
&,,color{blue}{=0}
end{align*}
answered Oct 26 at 20:29
Markus Scheuer
59.5k455142
add a comment |
up vote
1
down vote
up vote
1
down vote
Your answer is correct. The residue of a function is the coefficient of $z^{-1}$ of the Laurent series expansion of the function.
We obtain
begin{align*}
color{blue}{[z^{-1}]}&color{blue}{frac{10z^4-10sin z}{z^3}}\
&=[z^{-1}]left(10z-frac{10}{z^3}left(z-frac{z^3}{3!}+frac{z^5}{5!}-cdotsright)right)\
&=-10[z^{-1}]left(frac{1}{z^2}-frac{1}{3!}+frac{z^2}{5!}-cdotsright)\
&,,color{blue}{=0}
end{align*}
answered Oct 26 at 20:29
Markus Scheuer
59.5k455142
Your answer is correct. The residue of a function is the coefficient of $z^{-1}$ of the Laurent series expansion of the function.
We obtain
begin{align*}
color{blue}{[z^{-1}]}&color{blue}{frac{10z^4-10sin z}{z^3}}\
&=[z^{-1}]left(10z-frac{10}{z^3}left(z-frac{z^3}{3!}+frac{z^5}{5!}-cdotsright)right)\
&=-10[z^{-1}]left(frac{1}{z^2}-frac{1}{3!}+frac{z^2}{5!}-cdotsright)\
&,,color{blue}{=0}
end{align*}
answered Oct 26 at 20:29
Markus Scheuer
59.5k455142
answered Oct 26 at 20:29
Markus Scheuer
59.5k455142
answered Oct 26 at 20:29
Markus Scheuer
59.5k455142
answered Oct 26 at 20:29
Markus Scheuer
59.5k455142
59.5k455142
add a comment |
add a comment |
up vote
0
down vote
One way to work with sines and cosines in this problem is to reckon with their power series expansions found in many places, for instance here. You might also note that an even function has only even powers in its Laurent series whereas the residue comes from the $(-1)$ power, therefore even functions will have a residue of zero. Hmm, can you see why I would mention that last fact?
answered Oct 26 at 0:22
Oscar Lanzi
11.9k11936
add a comment |
up vote
0
down vote
One way to work with sines and cosines in this problem is to reckon with their power series expansions found in many places, for instance here. You might also note that an even function has only even powers in its Laurent series whereas the residue comes from the $(-1)$ power, therefore even functions will have a residue of zero. Hmm, can you see why I would mention that last fact?
answered Oct 26 at 0:22
Oscar Lanzi
11.9k11936
add a comment |
up vote
0
down vote
up vote
0
down vote
One way to work with sines and cosines in this problem is to reckon with their power series expansions found in many places, for instance here. You might also note that an even function has only even powers in its Laurent series whereas the residue comes from the $(-1)$ power, therefore even functions will have a residue of zero. Hmm, can you see why I would mention that last fact?
answered Oct 26 at 0:22
Oscar Lanzi
11.9k11936
One way to work with sines and cosines in this problem is to reckon with their power series expansions found in many places, for instance here. You might also note that an even function has only even powers in its Laurent series whereas the residue comes from the $(-1)$ power, therefore even functions will have a residue of zero. Hmm, can you see why I would mention that last fact?
answered Oct 26 at 0:22
Oscar Lanzi
11.9k11936
answered Oct 26 at 0:22
Oscar Lanzi
11.9k11936
answered Oct 26 at 0:22
Oscar Lanzi
11.9k11936
answered Oct 26 at 0:22
Oscar Lanzi
11.9k11936
11.9k11936
add a comment |
add a comment |
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2971408%2fcalculating-the-residue-of-frac10z4-10-sinzz3-z0-0%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Can you show us the pathway you chose so we can compare it with what we would do?
– Oscar Lanzi
Oct 26 at 0:15