How to find an analytic extension of a function?
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For example, the function is $f(z)=frac{sin(z-i)}{z^2+1}$. Could you also list a general procedure to find an analytic extension?
Kavi says the $f(z)$ above can't be extended. Under what circumstances we can find an analytic extension of a function?
Since $f(z)$ can't be extended, what about $g(z)=frac{1}{e^{2z}-1}$? Or you may like to use another (nontrivial) example to illustrate that.
complex-analysis
asked Nov 15 at 7:35
user398843
536215
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up vote
0
down vote
favorite
For example, the function is $f(z)=frac{sin(z-i)}{z^2+1}$. Could you also list a general procedure to find an analytic extension?
Kavi says the $f(z)$ above can't be extended. Under what circumstances we can find an analytic extension of a function?
Since $f(z)$ can't be extended, what about $g(z)=frac{1}{e^{2z}-1}$? Or you may like to use another (nontrivial) example to illustrate that.
complex-analysis
asked Nov 15 at 7:35
user398843
536215
1
The function is analytic on $mathbb C setminus {-i}$ it cannot be extended beyond this.
– Kavi Rama Murthy
Nov 15 at 7:36
@KaviRamaMurthy Under what circumstances we can find an analytic extension?
– user398843
Nov 15 at 7:37
2
The question is too general. Finding analytic extension can be very hard and tricky.
– Kavi Rama Murthy
Nov 15 at 7:47
@KaviRamaMurthy What about my second toy function $g$? Can we find an analytic extension for that?
– user398843
Nov 15 at 7:48
Almost like $f$, but with a infinite number of poles. Both of your examples are trivial. Interesting example: the zeta function defined as the sum of the Dirichlet series.
– Martín-Blas Pérez Pinilla
Nov 15 at 9:39
|
show 2 more comments
up vote
0
down vote
favorite
up vote
0
down vote
favorite
For example, the function is $f(z)=frac{sin(z-i)}{z^2+1}$. Could you also list a general procedure to find an analytic extension?
Kavi says the $f(z)$ above can't be extended. Under what circumstances we can find an analytic extension of a function?
Since $f(z)$ can't be extended, what about $g(z)=frac{1}{e^{2z}-1}$? Or you may like to use another (nontrivial) example to illustrate that.
complex-analysis
asked Nov 15 at 7:35
user398843
536215
For example, the function is $f(z)=frac{sin(z-i)}{z^2+1}$. Could you also list a general procedure to find an analytic extension?
Kavi says the $f(z)$ above can't be extended. Under what circumstances we can find an analytic extension of a function?
Since $f(z)$ can't be extended, what about $g(z)=frac{1}{e^{2z}-1}$? Or you may like to use another (nontrivial) example to illustrate that.
complex-analysis
complex-analysis
asked Nov 15 at 7:35
user398843
536215
asked Nov 15 at 7:35
user398843
536215
edited Nov 15 at 7:41
asked Nov 15 at 7:35
user398843
536215
asked Nov 15 at 7:35
user398843
536215
asked Nov 15 at 7:35
user398843
536215
536215
1
The function is analytic on $mathbb C setminus {-i}$ it cannot be extended beyond this.
– Kavi Rama Murthy
Nov 15 at 7:36
@KaviRamaMurthy Under what circumstances we can find an analytic extension?
– user398843
Nov 15 at 7:37
2
The question is too general. Finding analytic extension can be very hard and tricky.
– Kavi Rama Murthy
Nov 15 at 7:47
@KaviRamaMurthy What about my second toy function $g$? Can we find an analytic extension for that?
– user398843
Nov 15 at 7:48
Almost like $f$, but with a infinite number of poles. Both of your examples are trivial. Interesting example: the zeta function defined as the sum of the Dirichlet series.
– Martín-Blas Pérez Pinilla
Nov 15 at 9:39
|
show 2 more comments
1
The function is analytic on $mathbb C setminus {-i}$ it cannot be extended beyond this.
– Kavi Rama Murthy
Nov 15 at 7:36
@KaviRamaMurthy Under what circumstances we can find an analytic extension?
– user398843
Nov 15 at 7:37
2
The question is too general. Finding analytic extension can be very hard and tricky.
– Kavi Rama Murthy
Nov 15 at 7:47
@KaviRamaMurthy What about my second toy function $g$? Can we find an analytic extension for that?
– user398843
Nov 15 at 7:48
Almost like $f$, but with a infinite number of poles. Both of your examples are trivial. Interesting example: the zeta function defined as the sum of the Dirichlet series.
– Martín-Blas Pérez Pinilla
Nov 15 at 9:39
1
1
The function is analytic on $mathbb C setminus {-i}$ it cannot be extended beyond this.
– Kavi Rama Murthy
Nov 15 at 7:36
The function is analytic on $mathbb C setminus {-i}$ it cannot be extended beyond this.
– Kavi Rama Murthy
Nov 15 at 7:36
@KaviRamaMurthy Under what circumstances we can find an analytic extension?
– user398843
Nov 15 at 7:37
@KaviRamaMurthy Under what circumstances we can find an analytic extension?
– user398843
Nov 15 at 7:37
2
2
The question is too general. Finding analytic extension can be very hard and tricky.
– Kavi Rama Murthy
Nov 15 at 7:47
The question is too general. Finding analytic extension can be very hard and tricky.
– Kavi Rama Murthy
Nov 15 at 7:47
@KaviRamaMurthy What about my second toy function $g$? Can we find an analytic extension for that?
– user398843
Nov 15 at 7:48
@KaviRamaMurthy What about my second toy function $g$? Can we find an analytic extension for that?
– user398843
Nov 15 at 7:48
Almost like $f$, but with a infinite number of poles. Both of your examples are trivial. Interesting example: the zeta function defined as the sum of the Dirichlet series.
– Martín-Blas Pérez Pinilla
Nov 15 at 9:39
Almost like $f$, but with a infinite number of poles. Both of your examples are trivial. Interesting example: the zeta function defined as the sum of the Dirichlet series.
– Martín-Blas Pérez Pinilla
Nov 15 at 9:39
|
show 2 more comments
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1
The function is analytic on $mathbb C setminus {-i}$ it cannot be extended beyond this.
– Kavi Rama Murthy
Nov 15 at 7:36
@KaviRamaMurthy Under what circumstances we can find an analytic extension?
– user398843
Nov 15 at 7:37
2
The question is too general. Finding analytic extension can be very hard and tricky.
– Kavi Rama Murthy
Nov 15 at 7:47
@KaviRamaMurthy What about my second toy function $g$? Can we find an analytic extension for that?
– user398843
Nov 15 at 7:48
Almost like $f$, but with a infinite number of poles. Both of your examples are trivial. Interesting example: the zeta function defined as the sum of the Dirichlet series.
– Martín-Blas Pérez Pinilla
Nov 15 at 9:39