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.










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  • 1




    The function is analytic on $mathbb C setminus {-i}$ it cannot be extended beyond this.
    – Kavi Rama Murthy
    yesterday












  • @KaviRamaMurthy Under what circumstances we can find an analytic extension?
    – user398843
    yesterday






  • 2




    The question is too general. Finding analytic extension can be very hard and tricky.
    – Kavi Rama Murthy
    yesterday










  • @KaviRamaMurthy What about my second toy function $g$? Can we find an analytic extension for that?
    – user398843
    yesterday












  • 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
    yesterday

















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.










share|cite|improve this question




















  • 1




    The function is analytic on $mathbb C setminus {-i}$ it cannot be extended beyond this.
    – Kavi Rama Murthy
    yesterday












  • @KaviRamaMurthy Under what circumstances we can find an analytic extension?
    – user398843
    yesterday






  • 2




    The question is too general. Finding analytic extension can be very hard and tricky.
    – Kavi Rama Murthy
    yesterday










  • @KaviRamaMurthy What about my second toy function $g$? Can we find an analytic extension for that?
    – user398843
    yesterday












  • 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
    yesterday















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.










share|cite|improve this question















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






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share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited yesterday

























asked yesterday









user398843

529215




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  • 1




    The function is analytic on $mathbb C setminus {-i}$ it cannot be extended beyond this.
    – Kavi Rama Murthy
    yesterday












  • @KaviRamaMurthy Under what circumstances we can find an analytic extension?
    – user398843
    yesterday






  • 2




    The question is too general. Finding analytic extension can be very hard and tricky.
    – Kavi Rama Murthy
    yesterday










  • @KaviRamaMurthy What about my second toy function $g$? Can we find an analytic extension for that?
    – user398843
    yesterday












  • 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
    yesterday
















  • 1




    The function is analytic on $mathbb C setminus {-i}$ it cannot be extended beyond this.
    – Kavi Rama Murthy
    yesterday












  • @KaviRamaMurthy Under what circumstances we can find an analytic extension?
    – user398843
    yesterday






  • 2




    The question is too general. Finding analytic extension can be very hard and tricky.
    – Kavi Rama Murthy
    yesterday










  • @KaviRamaMurthy What about my second toy function $g$? Can we find an analytic extension for that?
    – user398843
    yesterday












  • 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
    yesterday










1




1




The function is analytic on $mathbb C setminus {-i}$ it cannot be extended beyond this.
– Kavi Rama Murthy
yesterday






The function is analytic on $mathbb C setminus {-i}$ it cannot be extended beyond this.
– Kavi Rama Murthy
yesterday














@KaviRamaMurthy Under what circumstances we can find an analytic extension?
– user398843
yesterday




@KaviRamaMurthy Under what circumstances we can find an analytic extension?
– user398843
yesterday




2




2




The question is too general. Finding analytic extension can be very hard and tricky.
– Kavi Rama Murthy
yesterday




The question is too general. Finding analytic extension can be very hard and tricky.
– Kavi Rama Murthy
yesterday












@KaviRamaMurthy What about my second toy function $g$? Can we find an analytic extension for that?
– user398843
yesterday






@KaviRamaMurthy What about my second toy function $g$? Can we find an analytic extension for that?
– user398843
yesterday














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
yesterday






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
yesterday

















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