How do i represent an infinite series as an integral? (complex numbers)

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My calculus book has nothing on complex numbers so here i am.



$$ e^z = sum_0^infty fracz^nn! $$



Given the above function as an example , can someone take me through the steps as to how i would represent it as an integral? and perhaps offer an extra example where i don't know the function , just the sum?



Thank you very much for your time and help.






sequences-and-series complex-numbers







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  • Are you asking how to convert the sum into an integral?
    – Tom Himler
    Sep 1 at 3:46










  • yes , if possible, i am showing what this infinite sum is (e^z) but my ideal example would be i am given an infinite sum , and i convert that sum into an integral
    – Victor Orta
    Sep 1 at 3:48














up vote
0
down vote

favorite












My calculus book has nothing on complex numbers so here i am.



$$ e^z = sum_0^infty fracz^nn! $$



Given the above function as an example , can someone take me through the steps as to how i would represent it as an integral? and perhaps offer an extra example where i don't know the function , just the sum?



Thank you very much for your time and help.






sequences-and-series complex-numbers







share|cite|improve this question





















  • Are you asking how to convert the sum into an integral?
    – Tom Himler
    Sep 1 at 3:46










  • yes , if possible, i am showing what this infinite sum is (e^z) but my ideal example would be i am given an infinite sum , and i convert that sum into an integral
    – Victor Orta
    Sep 1 at 3:48












up vote
0
down vote

favorite









up vote
0
down vote

favorite











My calculus book has nothing on complex numbers so here i am.



$$ e^z = sum_0^infty fracz^nn! $$



Given the above function as an example , can someone take me through the steps as to how i would represent it as an integral? and perhaps offer an extra example where i don't know the function , just the sum?



Thank you very much for your time and help.






sequences-and-series complex-numbers







share|cite|improve this question













My calculus book has nothing on complex numbers so here i am.



$$ e^z = sum_0^infty fracz^nn! $$



Given the above function as an example , can someone take me through the steps as to how i would represent it as an integral? and perhaps offer an extra example where i don't know the function , just the sum?



Thank you very much for your time and help.






sequences-and-series complex-numbers




sequences-and-series complex-numbers






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked Sep 1 at 3:37









Victor Orta

545




545











  • Are you asking how to convert the sum into an integral?
    – Tom Himler
    Sep 1 at 3:46










  • yes , if possible, i am showing what this infinite sum is (e^z) but my ideal example would be i am given an infinite sum , and i convert that sum into an integral
    – Victor Orta
    Sep 1 at 3:48
















  • Are you asking how to convert the sum into an integral?
    – Tom Himler
    Sep 1 at 3:46










  • yes , if possible, i am showing what this infinite sum is (e^z) but my ideal example would be i am given an infinite sum , and i convert that sum into an integral
    – Victor Orta
    Sep 1 at 3:48















Are you asking how to convert the sum into an integral?
– Tom Himler
Sep 1 at 3:46




Are you asking how to convert the sum into an integral?
– Tom Himler
Sep 1 at 3:46












yes , if possible, i am showing what this infinite sum is (e^z) but my ideal example would be i am given an infinite sum , and i convert that sum into an integral
– Victor Orta
Sep 1 at 3:48




yes , if possible, i am showing what this infinite sum is (e^z) but my ideal example would be i am given an infinite sum , and i convert that sum into an integral
– Victor Orta
Sep 1 at 3:48










1 Answer
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I assume you are familiar with Taylor's Theorem and Taylor's Expansion of Analytic function. See Analytic functions. Now the term $f^(n)(z)$ can be expressed by the help of Cauchy Integral Formula for $n^th$ derivative as, $$a_nn!=f^(n)(a)=fracn!2pi ioint_gamma fracf(z)(z-a)^n+1dz$$ $forall n=0,1,2...$. Notations are defined in the links.






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    up vote
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    I assume you are familiar with Taylor's Theorem and Taylor's Expansion of Analytic function. See Analytic functions. Now the term $f^(n)(z)$ can be expressed by the help of Cauchy Integral Formula for $n^th$ derivative as, $$a_nn!=f^(n)(a)=fracn!2pi ioint_gamma fracf(z)(z-a)^n+1dz$$ $forall n=0,1,2...$. Notations are defined in the links.






    share|cite|improve this answer
























      up vote
      2
      down vote













      I assume you are familiar with Taylor's Theorem and Taylor's Expansion of Analytic function. See Analytic functions. Now the term $f^(n)(z)$ can be expressed by the help of Cauchy Integral Formula for $n^th$ derivative as, $$a_nn!=f^(n)(a)=fracn!2pi ioint_gamma fracf(z)(z-a)^n+1dz$$ $forall n=0,1,2...$. Notations are defined in the links.






      share|cite|improve this answer






















        up vote
        2
        down vote










        up vote
        2
        down vote









        I assume you are familiar with Taylor's Theorem and Taylor's Expansion of Analytic function. See Analytic functions. Now the term $f^(n)(z)$ can be expressed by the help of Cauchy Integral Formula for $n^th$ derivative as, $$a_nn!=f^(n)(a)=fracn!2pi ioint_gamma fracf(z)(z-a)^n+1dz$$ $forall n=0,1,2...$. Notations are defined in the links.






        share|cite|improve this answer












        I assume you are familiar with Taylor's Theorem and Taylor's Expansion of Analytic function. See Analytic functions. Now the term $f^(n)(z)$ can be expressed by the help of Cauchy Integral Formula for $n^th$ derivative as, $$a_nn!=f^(n)(a)=fracn!2pi ioint_gamma fracf(z)(z-a)^n+1dz$$ $forall n=0,1,2...$. Notations are defined in the links.







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        answered Sep 1 at 3:49









        Sujit Bhattacharyya

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