Mixing
![Determine the matrix A that represents T in the canonical form of R2[x] in the start and in the end](https://blogger.googleusercontent.com/img/b/R29vZ2xl/AVvXsEhZCTB4-bb-s-32SAm6ytjzFOU06cH9o8q-a_wq_UnH6lcd8hvws23CmBWHM6g256LzTX9yK1EiCCzTC1NSgtxlNk_J9hdr9bA3tD0Nqntbl521j4bLAm_uXPANuWLBhcCKTKzns2gaUodx/s72-c/1.jpg)
Residue of $frac1(z^2-1)^3$ at the singularities.
Clash Royale CLAN TAG#URR8PPP
up vote
-1
down vote
favorite
Find the residue of $$frac1(z^2-1)^3$$ at the singularities.
To find which order of pole it is I tried to look to the order of zeros of $(z^2-1)^3$ deriving the function seems to big the hard way so I tried to look at the Laurent series $$frac1(z^2-1)^3=frac1z^6(1-frac1z^6)^3$$
Can I do such a thing?
complex-analysis residue-calculus
edited Jul 15 at 15:33
Nosrati
19.9k41644
asked Jul 15 at 14:43
newhere
759310
add a comment |Â
up vote
-1
down vote
favorite
Find the residue of $$frac1(z^2-1)^3$$ at the singularities.
To find which order of pole it is I tried to look to the order of zeros of $(z^2-1)^3$ deriving the function seems to big the hard way so I tried to look at the Laurent series $$frac1(z^2-1)^3=frac1z^6(1-frac1z^6)^3$$
Can I do such a thing?
complex-analysis residue-calculus
edited Jul 15 at 15:33
Nosrati
19.9k41644
asked Jul 15 at 14:43
newhere
759310
add a comment |Â
up vote
-1
down vote
favorite
up vote
-1
down vote
favorite
Find the residue of $$frac1(z^2-1)^3$$ at the singularities.
To find which order of pole it is I tried to look to the order of zeros of $(z^2-1)^3$ deriving the function seems to big the hard way so I tried to look at the Laurent series $$frac1(z^2-1)^3=frac1z^6(1-frac1z^6)^3$$
Can I do such a thing?
complex-analysis residue-calculus
edited Jul 15 at 15:33
Nosrati
19.9k41644
asked Jul 15 at 14:43
newhere
759310
Find the residue of $$frac1(z^2-1)^3$$ at the singularities.
To find which order of pole it is I tried to look to the order of zeros of $(z^2-1)^3$ deriving the function seems to big the hard way so I tried to look at the Laurent series $$frac1(z^2-1)^3=frac1z^6(1-frac1z^6)^3$$
Can I do such a thing?
complex-analysis residue-calculus
edited Jul 15 at 15:33
Nosrati
19.9k41644
asked Jul 15 at 14:43
newhere
759310
edited Jul 15 at 15:33
Nosrati
19.9k41644
edited Jul 15 at 15:33
Nosrati
19.9k41644
edited Jul 15 at 15:33
Nosrati
19.9k41644
19.9k41644
asked Jul 15 at 14:43
newhere
759310
asked Jul 15 at 14:43
newhere
759310
asked Jul 15 at 14:43
newhere
759310
759310
add a comment |Â
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
1
down vote
accepted
Big Hint: $z^2-1=(z+1)(z-1).$
answered Jul 15 at 15:04
Cameron Buie
83.5k771153
add a comment |Â
up vote
2
down vote
Note thatbeginalign(z+1)^-3&=frac12bigl((z+1)^-1bigr)''\&=frac12left(frac12-fracz-14+frac(z-1)^28-frac(z-1)^316+frac(z-1)^432-cdotsright)''\&=frac18-frac316(z-1)+frac316(z-1)^2-cdotsendalignThereforebeginalignfrac1(z^2-1)^3&=frac1(z-1)^3(z+1)^3\&=frac18(z-1)^-3-frac316(z-1)^-2+frac316(z-1)^-1+cdotsendalignand so$$operatornameResleft(frac1(z^2-1)^3,1right)=frac316.$$Can you compute the other residue now?
answered Jul 15 at 15:05
José Carlos Santos
114k1698177
add a comment |Â
2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
1
down vote
accepted
Big Hint: $z^2-1=(z+1)(z-1).$
answered Jul 15 at 15:04
Cameron Buie
83.5k771153
add a comment |Â
up vote
1
down vote
accepted
Big Hint: $z^2-1=(z+1)(z-1).$
answered Jul 15 at 15:04
Cameron Buie
83.5k771153
add a comment |Â
up vote
1
down vote
accepted
up vote
1
down vote
accepted
Big Hint: $z^2-1=(z+1)(z-1).$
answered Jul 15 at 15:04
Cameron Buie
83.5k771153
Big Hint: $z^2-1=(z+1)(z-1).$
answered Jul 15 at 15:04
Cameron Buie
83.5k771153
answered Jul 15 at 15:04
Cameron Buie
83.5k771153
answered Jul 15 at 15:04
Cameron Buie
83.5k771153
answered Jul 15 at 15:04
Cameron Buie
83.5k771153
83.5k771153
add a comment |Â
add a comment |Â
up vote
2
down vote
Note thatbeginalign(z+1)^-3&=frac12bigl((z+1)^-1bigr)''\&=frac12left(frac12-fracz-14+frac(z-1)^28-frac(z-1)^316+frac(z-1)^432-cdotsright)''\&=frac18-frac316(z-1)+frac316(z-1)^2-cdotsendalignThereforebeginalignfrac1(z^2-1)^3&=frac1(z-1)^3(z+1)^3\&=frac18(z-1)^-3-frac316(z-1)^-2+frac316(z-1)^-1+cdotsendalignand so$$operatornameResleft(frac1(z^2-1)^3,1right)=frac316.$$Can you compute the other residue now?
answered Jul 15 at 15:05
José Carlos Santos
114k1698177
add a comment |Â
up vote
2
down vote
Note thatbeginalign(z+1)^-3&=frac12bigl((z+1)^-1bigr)''\&=frac12left(frac12-fracz-14+frac(z-1)^28-frac(z-1)^316+frac(z-1)^432-cdotsright)''\&=frac18-frac316(z-1)+frac316(z-1)^2-cdotsendalignThereforebeginalignfrac1(z^2-1)^3&=frac1(z-1)^3(z+1)^3\&=frac18(z-1)^-3-frac316(z-1)^-2+frac316(z-1)^-1+cdotsendalignand so$$operatornameResleft(frac1(z^2-1)^3,1right)=frac316.$$Can you compute the other residue now?
answered Jul 15 at 15:05
José Carlos Santos
114k1698177
add a comment |Â
up vote
2
down vote
up vote
2
down vote
Note thatbeginalign(z+1)^-3&=frac12bigl((z+1)^-1bigr)''\&=frac12left(frac12-fracz-14+frac(z-1)^28-frac(z-1)^316+frac(z-1)^432-cdotsright)''\&=frac18-frac316(z-1)+frac316(z-1)^2-cdotsendalignThereforebeginalignfrac1(z^2-1)^3&=frac1(z-1)^3(z+1)^3\&=frac18(z-1)^-3-frac316(z-1)^-2+frac316(z-1)^-1+cdotsendalignand so$$operatornameResleft(frac1(z^2-1)^3,1right)=frac316.$$Can you compute the other residue now?
answered Jul 15 at 15:05
José Carlos Santos
114k1698177
Note thatbeginalign(z+1)^-3&=frac12bigl((z+1)^-1bigr)''\&=frac12left(frac12-fracz-14+frac(z-1)^28-frac(z-1)^316+frac(z-1)^432-cdotsright)''\&=frac18-frac316(z-1)+frac316(z-1)^2-cdotsendalignThereforebeginalignfrac1(z^2-1)^3&=frac1(z-1)^3(z+1)^3\&=frac18(z-1)^-3-frac316(z-1)^-2+frac316(z-1)^-1+cdotsendalignand so$$operatornameResleft(frac1(z^2-1)^3,1right)=frac316.$$Can you compute the other residue now?
answered Jul 15 at 15:05
José Carlos Santos
114k1698177
answered Jul 15 at 15:05
José Carlos Santos
114k1698177
answered Jul 15 at 15:05
José Carlos Santos
114k1698177
answered Jul 15 at 15:05
José Carlos Santos
114k1698177
114k1698177
add a comment |Â
add a comment |Â
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
StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2852587%2fresidue-of-frac1z2-13-at-the-singularities%23new-answer', 'question_page');
);
Post as a guest
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
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
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