Taylor series convergence of $f(z)=int^z_0 frac{zeta-sin(zeta)}{zeta^2+4} , dzeta$ at $a=2$.
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$$f(z)=int^z_0 frac{zeta-sin(zeta)}{zeta^2+4} , dzeta$$
I am supposed to find the convergence radius of its Taylor series at point $a=2$.
I can find the radius in simple cases by finding the distance from the nearest non-analytic point, but I am not sure how to handle the current situation. I believe that it is not necessary to actually find the series, also, integrating it seems to get a bit ugly. How can I figure out the points where the function fails to be analytic? I would really appreciate some suggestions.
complex-analysis taylor-expansion
edited Nov 26 at 16:27
Shaun
8,190113577
asked Nov 5 '13 at 1:53
Andres Traumann
62
add a comment |
up vote
1
down vote
favorite
$$f(z)=int^z_0 frac{zeta-sin(zeta)}{zeta^2+4} , dzeta$$
I am supposed to find the convergence radius of its Taylor series at point $a=2$.
I can find the radius in simple cases by finding the distance from the nearest non-analytic point, but I am not sure how to handle the current situation. I believe that it is not necessary to actually find the series, also, integrating it seems to get a bit ugly. How can I figure out the points where the function fails to be analytic? I would really appreciate some suggestions.
complex-analysis taylor-expansion
edited Nov 26 at 16:27
Shaun
8,190113577
asked Nov 5 '13 at 1:53
Andres Traumann
62
$f'(z) = frac{z-sinz}{z^2 + 4}$ (fundamental theorem of calc). I would say f(z) has poles at $pm$ 2i.
– Betty Mock
Nov 5 '13 at 3:02
add a comment |
up vote
1
down vote
favorite
up vote
1
down vote
favorite
$$f(z)=int^z_0 frac{zeta-sin(zeta)}{zeta^2+4} , dzeta$$
I am supposed to find the convergence radius of its Taylor series at point $a=2$.
I can find the radius in simple cases by finding the distance from the nearest non-analytic point, but I am not sure how to handle the current situation. I believe that it is not necessary to actually find the series, also, integrating it seems to get a bit ugly. How can I figure out the points where the function fails to be analytic? I would really appreciate some suggestions.
complex-analysis taylor-expansion
edited Nov 26 at 16:27
Shaun
8,190113577
asked Nov 5 '13 at 1:53
Andres Traumann
62
$$f(z)=int^z_0 frac{zeta-sin(zeta)}{zeta^2+4} , dzeta$$
I am supposed to find the convergence radius of its Taylor series at point $a=2$.
I can find the radius in simple cases by finding the distance from the nearest non-analytic point, but I am not sure how to handle the current situation. I believe that it is not necessary to actually find the series, also, integrating it seems to get a bit ugly. How can I figure out the points where the function fails to be analytic? I would really appreciate some suggestions.
complex-analysis taylor-expansion
complex-analysis taylor-expansion
edited Nov 26 at 16:27
Shaun
8,190113577
asked Nov 5 '13 at 1:53
Andres Traumann
62
edited Nov 26 at 16:27
Shaun
8,190113577
asked Nov 5 '13 at 1:53
Andres Traumann
62
edited Nov 26 at 16:27
Shaun
8,190113577
edited Nov 26 at 16:27
Shaun
8,190113577
edited Nov 26 at 16:27
Shaun
8,190113577
8,190113577
asked Nov 5 '13 at 1:53
Andres Traumann
62
asked Nov 5 '13 at 1:53
Andres Traumann
62
asked Nov 5 '13 at 1:53
Andres Traumann
62
62
$f'(z) = frac{z-sinz}{z^2 + 4}$ (fundamental theorem of calc). I would say f(z) has poles at $pm$ 2i.
– Betty Mock
Nov 5 '13 at 3:02
add a comment |
$f'(z) = frac{z-sinz}{z^2 + 4}$ (fundamental theorem of calc). I would say f(z) has poles at $pm$ 2i.
– Betty Mock
Nov 5 '13 at 3:02
$f'(z) = frac{z-sinz}{z^2 + 4}$ (fundamental theorem of calc). I would say f(z) has poles at $pm$ 2i.
– Betty Mock
Nov 5 '13 at 3:02
$f'(z) = frac{z-sinz}{z^2 + 4}$ (fundamental theorem of calc). I would say f(z) has poles at $pm$ 2i.
– Betty Mock
Nov 5 '13 at 3:02
add a comment |
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$f'(z) = frac{z-sinz}{z^2 + 4}$ (fundamental theorem of calc). I would say f(z) has poles at $pm$ 2i.
– Betty Mock
Nov 5 '13 at 3:02