Closed Form of the Real Portion of $f(n) = prod_{m=2}^{n-1} e^{pi i n/m}$












3












$begingroup$


I am wondering if it is possible to express an equation in closed form. I currently have:



$$f(n) = prod_{m=2}^{n-1} e^{pi i n/m}$$



Where $i$ is the $sqrt{-1}$, which I know it commonly represents but due to the finite product I figured I would add this note for clarity.



Context: I am trying to work towards a solution for Simplify Product of sines, yet so far I have only arrived here.



Some options I have been considering to try and find a closed form include cases such as I only actually care when $n$ is an odd number so discarding even cases is fine if that simplifies it. I am also fine with having the series start a $m=1$ instead of $m=2$ if that somehow simplifies the answer. The other potential saving grace is that I may only care about the real portion of the answer and not the imaginary portion if that simplifies things.



Sorry I am a computer scientist not a mathematician so please let me know if I should adjust the title or the wording of the question for clarity. Any leads or help of any form would be appreciated as I am currently lost.






exponential-function closed-form products







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  • $begingroup$
    Please give more context. Providing context not only assures that this is not simply copied from a homework assignment, but also allows answers to be better directed at where the problem lies and to be within the proper scope. Please avoid "I have no clue" questions. Defining keywords and trying a simpler, similar problem often helps.
    $endgroup$
    – robjohn♦
    Jan 17 at 3:15






  • 1




    $begingroup$
    Hi, I guess the context is I was trying to work a little more on this, math.stackexchange.com/questions/1689831/…. I'm not in school and this is not homework, I am just a computer scientist who likes to dabble in some of my free time. I was trying to work towards a closed form of the equation linked and so I was breaking it into parts and this is where I got stuck. Its something I look at every few months so even a nudge in any direction would be helpful.
    $endgroup$
    – cytinus
    Jan 17 at 3:22










  • $begingroup$
    Hint: $prodlimits_{m=2}^{n-1}e^{a_m}=e^{sumlimits_{m=2}^{n-1}a_m}$
    $endgroup$
    – robjohn♦
    Jan 17 at 3:25






  • 1




    $begingroup$
    It would be helpful to add that context to the question itself, rather than in a comment. I have undeleted my answer.
    $endgroup$
    – robjohn♦
    Jan 17 at 3:29


















3












$begingroup$


I am wondering if it is possible to express an equation in closed form. I currently have:



$$f(n) = prod_{m=2}^{n-1} e^{pi i n/m}$$



Where $i$ is the $sqrt{-1}$, which I know it commonly represents but due to the finite product I figured I would add this note for clarity.



Context: I am trying to work towards a solution for Simplify Product of sines, yet so far I have only arrived here.



Some options I have been considering to try and find a closed form include cases such as I only actually care when $n$ is an odd number so discarding even cases is fine if that simplifies it. I am also fine with having the series start a $m=1$ instead of $m=2$ if that somehow simplifies the answer. The other potential saving grace is that I may only care about the real portion of the answer and not the imaginary portion if that simplifies things.



Sorry I am a computer scientist not a mathematician so please let me know if I should adjust the title or the wording of the question for clarity. Any leads or help of any form would be appreciated as I am currently lost.






exponential-function closed-form products







share|cite|improve this question









New contributor




cytinus is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$












  • $begingroup$
    Please give more context. Providing context not only assures that this is not simply copied from a homework assignment, but also allows answers to be better directed at where the problem lies and to be within the proper scope. Please avoid "I have no clue" questions. Defining keywords and trying a simpler, similar problem often helps.
    $endgroup$
    – robjohn♦
    Jan 17 at 3:15






  • 1




    $begingroup$
    Hi, I guess the context is I was trying to work a little more on this, math.stackexchange.com/questions/1689831/…. I'm not in school and this is not homework, I am just a computer scientist who likes to dabble in some of my free time. I was trying to work towards a closed form of the equation linked and so I was breaking it into parts and this is where I got stuck. Its something I look at every few months so even a nudge in any direction would be helpful.
    $endgroup$
    – cytinus
    Jan 17 at 3:22










  • $begingroup$
    Hint: $prodlimits_{m=2}^{n-1}e^{a_m}=e^{sumlimits_{m=2}^{n-1}a_m}$
    $endgroup$
    – robjohn♦
    Jan 17 at 3:25






  • 1




    $begingroup$
    It would be helpful to add that context to the question itself, rather than in a comment. I have undeleted my answer.
    $endgroup$
    – robjohn♦
    Jan 17 at 3:29
















3












3








3





$begingroup$


I am wondering if it is possible to express an equation in closed form. I currently have:



$$f(n) = prod_{m=2}^{n-1} e^{pi i n/m}$$



Where $i$ is the $sqrt{-1}$, which I know it commonly represents but due to the finite product I figured I would add this note for clarity.



Context: I am trying to work towards a solution for Simplify Product of sines, yet so far I have only arrived here.



Some options I have been considering to try and find a closed form include cases such as I only actually care when $n$ is an odd number so discarding even cases is fine if that simplifies it. I am also fine with having the series start a $m=1$ instead of $m=2$ if that somehow simplifies the answer. The other potential saving grace is that I may only care about the real portion of the answer and not the imaginary portion if that simplifies things.



Sorry I am a computer scientist not a mathematician so please let me know if I should adjust the title or the wording of the question for clarity. Any leads or help of any form would be appreciated as I am currently lost.






exponential-function closed-form products







share|cite|improve this question









New contributor




cytinus is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$




I am wondering if it is possible to express an equation in closed form. I currently have:



$$f(n) = prod_{m=2}^{n-1} e^{pi i n/m}$$



Where $i$ is the $sqrt{-1}$, which I know it commonly represents but due to the finite product I figured I would add this note for clarity.



Context: I am trying to work towards a solution for Simplify Product of sines, yet so far I have only arrived here.



Some options I have been considering to try and find a closed form include cases such as I only actually care when $n$ is an odd number so discarding even cases is fine if that simplifies it. I am also fine with having the series start a $m=1$ instead of $m=2$ if that somehow simplifies the answer. The other potential saving grace is that I may only care about the real portion of the answer and not the imaginary portion if that simplifies things.



Sorry I am a computer scientist not a mathematician so please let me know if I should adjust the title or the wording of the question for clarity. Any leads or help of any form would be appreciated as I am currently lost.






exponential-function closed-form products




exponential-function closed-form products






share|cite|improve this question









New contributor




cytinus is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question









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share|cite|improve this question




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edited Jan 17 at 8:26









Asaf Karagila♦

303k32429760




303k32429760






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asked Jan 17 at 2:44









cytinuscytinus

1164




1164




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New contributor





cytinus is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






cytinus is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.












  • $begingroup$
    Please give more context. Providing context not only assures that this is not simply copied from a homework assignment, but also allows answers to be better directed at where the problem lies and to be within the proper scope. Please avoid "I have no clue" questions. Defining keywords and trying a simpler, similar problem often helps.
    $endgroup$
    – robjohn♦
    Jan 17 at 3:15






  • 1




    $begingroup$
    Hi, I guess the context is I was trying to work a little more on this, math.stackexchange.com/questions/1689831/…. I'm not in school and this is not homework, I am just a computer scientist who likes to dabble in some of my free time. I was trying to work towards a closed form of the equation linked and so I was breaking it into parts and this is where I got stuck. Its something I look at every few months so even a nudge in any direction would be helpful.
    $endgroup$
    – cytinus
    Jan 17 at 3:22










  • $begingroup$
    Hint: $prodlimits_{m=2}^{n-1}e^{a_m}=e^{sumlimits_{m=2}^{n-1}a_m}$
    $endgroup$
    – robjohn♦
    Jan 17 at 3:25






  • 1




    $begingroup$
    It would be helpful to add that context to the question itself, rather than in a comment. I have undeleted my answer.
    $endgroup$
    – robjohn♦
    Jan 17 at 3:29




















  • $begingroup$
    Please give more context. Providing context not only assures that this is not simply copied from a homework assignment, but also allows answers to be better directed at where the problem lies and to be within the proper scope. Please avoid "I have no clue" questions. Defining keywords and trying a simpler, similar problem often helps.
    $endgroup$
    – robjohn♦
    Jan 17 at 3:15






  • 1




    $begingroup$
    Hi, I guess the context is I was trying to work a little more on this, math.stackexchange.com/questions/1689831/…. I'm not in school and this is not homework, I am just a computer scientist who likes to dabble in some of my free time. I was trying to work towards a closed form of the equation linked and so I was breaking it into parts and this is where I got stuck. Its something I look at every few months so even a nudge in any direction would be helpful.
    $endgroup$
    – cytinus
    Jan 17 at 3:22










  • $begingroup$
    Hint: $prodlimits_{m=2}^{n-1}e^{a_m}=e^{sumlimits_{m=2}^{n-1}a_m}$
    $endgroup$
    – robjohn♦
    Jan 17 at 3:25






  • 1




    $begingroup$
    It would be helpful to add that context to the question itself, rather than in a comment. I have undeleted my answer.
    $endgroup$
    – robjohn♦
    Jan 17 at 3:29


















$begingroup$
Please give more context. Providing context not only assures that this is not simply copied from a homework assignment, but also allows answers to be better directed at where the problem lies and to be within the proper scope. Please avoid "I have no clue" questions. Defining keywords and trying a simpler, similar problem often helps.
$endgroup$
– robjohn♦
Jan 17 at 3:15




$begingroup$
Please give more context. Providing context not only assures that this is not simply copied from a homework assignment, but also allows answers to be better directed at where the problem lies and to be within the proper scope. Please avoid "I have no clue" questions. Defining keywords and trying a simpler, similar problem often helps.
$endgroup$
– robjohn♦
Jan 17 at 3:15




1




1




$begingroup$
Hi, I guess the context is I was trying to work a little more on this, math.stackexchange.com/questions/1689831/…. I'm not in school and this is not homework, I am just a computer scientist who likes to dabble in some of my free time. I was trying to work towards a closed form of the equation linked and so I was breaking it into parts and this is where I got stuck. Its something I look at every few months so even a nudge in any direction would be helpful.
$endgroup$
– cytinus
Jan 17 at 3:22




$begingroup$
Hi, I guess the context is I was trying to work a little more on this, math.stackexchange.com/questions/1689831/…. I'm not in school and this is not homework, I am just a computer scientist who likes to dabble in some of my free time. I was trying to work towards a closed form of the equation linked and so I was breaking it into parts and this is where I got stuck. Its something I look at every few months so even a nudge in any direction would be helpful.
$endgroup$
– cytinus
Jan 17 at 3:22












$begingroup$
Hint: $prodlimits_{m=2}^{n-1}e^{a_m}=e^{sumlimits_{m=2}^{n-1}a_m}$
$endgroup$
– robjohn♦
Jan 17 at 3:25




$begingroup$
Hint: $prodlimits_{m=2}^{n-1}e^{a_m}=e^{sumlimits_{m=2}^{n-1}a_m}$
$endgroup$
– robjohn♦
Jan 17 at 3:25




1




1




$begingroup$
It would be helpful to add that context to the question itself, rather than in a comment. I have undeleted my answer.
$endgroup$
– robjohn♦
Jan 17 at 3:29






$begingroup$
It would be helpful to add that context to the question itself, rather than in a comment. I have undeleted my answer.
$endgroup$
– robjohn♦
Jan 17 at 3:29












2 Answers
2






active

oldest

votes


















6












$begingroup$

$$
begin{align}
prod_{m=2}^{n-1}e^{frac{pi in}m}
&=e^{pi in(H_{n-1}-1)}\
&=(-1)^ne^{pi inH_{n-1}}\[9pt]
&=(-1)^{n-1}e^{pi inH_n}tag1
end{align}
$$
where $H_n$ is the $n^text{th}$ Harmonic Number.
$$
H_nsimlog(n)+gamma+frac1{2n}-frac1{12n^2}+frac1{120n^4}-frac1{252n^6}+frac1{240n^8}-frac1{132n^{10}}tag2
$$
and $gamma$ is the Euler-Mascheroni constant.



The real portion of $(1)$ is
$$
operatorname{Re}left(prod_{m=2}^{n-1}e^{frac{pi in}m}right)=(-1)^{n-1}cosleft(pi nH_nright)tag3
$$






share|cite|improve this answer











$endgroup$





















    4












    $begingroup$

    $$begin{align}
    f(n)=&prod_{m=2}^{n-1}expfrac{ipi n}{m}\
    &=expleft[sum_{m=2}^{n-1}frac{ipi n}{m}right]\
    &=expleft[ipi nsum_{m=2}^{n-1}frac{1}{m}right]\
    end{align}$$
    Recalling the definition of the harmonic numbers:
    $$H_n=sum_{m=1}^nfrac1m$$
    We have that
    $$f(n)=expleft[ipi n(H_{n-1}-1)right]$$
    Then using $e^{itheta}=costheta+isintheta$,
    $$f(n)=cosleft[pi n(H_{n-1}-1)right]+isinleft[pi n(H_{n-1}-1)right]$$
    So
    $$text{Re}f(n)=cosleft[pi n(H_{n-1}-1)right]$$






    share|cite|improve this answer











    $endgroup$









    • 1




      $begingroup$
      Why is $expleft[ipi n(H_{n-1}-1)right]=exp(ipi n)exp(H_{n-1}-1)$?
      $endgroup$
      – greelious
      Jan 17 at 3:29










    • $begingroup$
      @greelious because I messed up... Thanks! :)
      $endgroup$
      – clathratus
      Jan 17 at 3:30










    • $begingroup$
      I think it is still not correct: $expleft[ipi n(H_{n-1}-1)right]=exp(i pi n H_{n-1})exp(-i pi n)=exp(i pi n H_{n-1})(-1)^n$
      $endgroup$
      – greelious
      Jan 17 at 3:37












    • $begingroup$
      @greelious okay look now
      $endgroup$
      – clathratus
      Jan 17 at 3:41










    • $begingroup$
      Looks good now.
      $endgroup$
      – greelious
      Jan 17 at 3:45











    Your Answer





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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    6












    $begingroup$

    $$
    begin{align}
    prod_{m=2}^{n-1}e^{frac{pi in}m}
    &=e^{pi in(H_{n-1}-1)}\
    &=(-1)^ne^{pi inH_{n-1}}\[9pt]
    &=(-1)^{n-1}e^{pi inH_n}tag1
    end{align}
    $$
    where $H_n$ is the $n^text{th}$ Harmonic Number.
    $$
    H_nsimlog(n)+gamma+frac1{2n}-frac1{12n^2}+frac1{120n^4}-frac1{252n^6}+frac1{240n^8}-frac1{132n^{10}}tag2
    $$
    and $gamma$ is the Euler-Mascheroni constant.



    The real portion of $(1)$ is
    $$
    operatorname{Re}left(prod_{m=2}^{n-1}e^{frac{pi in}m}right)=(-1)^{n-1}cosleft(pi nH_nright)tag3
    $$






    share|cite|improve this answer











    $endgroup$


















      6












      $begingroup$

      $$
      begin{align}
      prod_{m=2}^{n-1}e^{frac{pi in}m}
      &=e^{pi in(H_{n-1}-1)}\
      &=(-1)^ne^{pi inH_{n-1}}\[9pt]
      &=(-1)^{n-1}e^{pi inH_n}tag1
      end{align}
      $$
      where $H_n$ is the $n^text{th}$ Harmonic Number.
      $$
      H_nsimlog(n)+gamma+frac1{2n}-frac1{12n^2}+frac1{120n^4}-frac1{252n^6}+frac1{240n^8}-frac1{132n^{10}}tag2
      $$
      and $gamma$ is the Euler-Mascheroni constant.



      The real portion of $(1)$ is
      $$
      operatorname{Re}left(prod_{m=2}^{n-1}e^{frac{pi in}m}right)=(-1)^{n-1}cosleft(pi nH_nright)tag3
      $$






      share|cite|improve this answer











      $endgroup$
















        6












        6








        6





        $begingroup$

        $$
        begin{align}
        prod_{m=2}^{n-1}e^{frac{pi in}m}
        &=e^{pi in(H_{n-1}-1)}\
        &=(-1)^ne^{pi inH_{n-1}}\[9pt]
        &=(-1)^{n-1}e^{pi inH_n}tag1
        end{align}
        $$
        where $H_n$ is the $n^text{th}$ Harmonic Number.
        $$
        H_nsimlog(n)+gamma+frac1{2n}-frac1{12n^2}+frac1{120n^4}-frac1{252n^6}+frac1{240n^8}-frac1{132n^{10}}tag2
        $$
        and $gamma$ is the Euler-Mascheroni constant.



        The real portion of $(1)$ is
        $$
        operatorname{Re}left(prod_{m=2}^{n-1}e^{frac{pi in}m}right)=(-1)^{n-1}cosleft(pi nH_nright)tag3
        $$






        share|cite|improve this answer











        $endgroup$



        $$
        begin{align}
        prod_{m=2}^{n-1}e^{frac{pi in}m}
        &=e^{pi in(H_{n-1}-1)}\
        &=(-1)^ne^{pi inH_{n-1}}\[9pt]
        &=(-1)^{n-1}e^{pi inH_n}tag1
        end{align}
        $$
        where $H_n$ is the $n^text{th}$ Harmonic Number.
        $$
        H_nsimlog(n)+gamma+frac1{2n}-frac1{12n^2}+frac1{120n^4}-frac1{252n^6}+frac1{240n^8}-frac1{132n^{10}}tag2
        $$
        and $gamma$ is the Euler-Mascheroni constant.



        The real portion of $(1)$ is
        $$
        operatorname{Re}left(prod_{m=2}^{n-1}e^{frac{pi in}m}right)=(-1)^{n-1}cosleft(pi nH_nright)tag3
        $$







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Jan 17 at 3:27

























        answered Jan 17 at 2:58









        robjohn♦robjohn

        266k27305626




        266k27305626























            4












            $begingroup$

            $$begin{align}
            f(n)=&prod_{m=2}^{n-1}expfrac{ipi n}{m}\
            &=expleft[sum_{m=2}^{n-1}frac{ipi n}{m}right]\
            &=expleft[ipi nsum_{m=2}^{n-1}frac{1}{m}right]\
            end{align}$$
            Recalling the definition of the harmonic numbers:
            $$H_n=sum_{m=1}^nfrac1m$$
            We have that
            $$f(n)=expleft[ipi n(H_{n-1}-1)right]$$
            Then using $e^{itheta}=costheta+isintheta$,
            $$f(n)=cosleft[pi n(H_{n-1}-1)right]+isinleft[pi n(H_{n-1}-1)right]$$
            So
            $$text{Re}f(n)=cosleft[pi n(H_{n-1}-1)right]$$






            share|cite|improve this answer











            $endgroup$









            • 1




              $begingroup$
              Why is $expleft[ipi n(H_{n-1}-1)right]=exp(ipi n)exp(H_{n-1}-1)$?
              $endgroup$
              – greelious
              Jan 17 at 3:29










            • $begingroup$
              @greelious because I messed up... Thanks! :)
              $endgroup$
              – clathratus
              Jan 17 at 3:30










            • $begingroup$
              I think it is still not correct: $expleft[ipi n(H_{n-1}-1)right]=exp(i pi n H_{n-1})exp(-i pi n)=exp(i pi n H_{n-1})(-1)^n$
              $endgroup$
              – greelious
              Jan 17 at 3:37












            • $begingroup$
              @greelious okay look now
              $endgroup$
              – clathratus
              Jan 17 at 3:41










            • $begingroup$
              Looks good now.
              $endgroup$
              – greelious
              Jan 17 at 3:45
















            4












            $begingroup$

            $$begin{align}
            f(n)=&prod_{m=2}^{n-1}expfrac{ipi n}{m}\
            &=expleft[sum_{m=2}^{n-1}frac{ipi n}{m}right]\
            &=expleft[ipi nsum_{m=2}^{n-1}frac{1}{m}right]\
            end{align}$$
            Recalling the definition of the harmonic numbers:
            $$H_n=sum_{m=1}^nfrac1m$$
            We have that
            $$f(n)=expleft[ipi n(H_{n-1}-1)right]$$
            Then using $e^{itheta}=costheta+isintheta$,
            $$f(n)=cosleft[pi n(H_{n-1}-1)right]+isinleft[pi n(H_{n-1}-1)right]$$
            So
            $$text{Re}f(n)=cosleft[pi n(H_{n-1}-1)right]$$






            share|cite|improve this answer











            $endgroup$









            • 1




              $begingroup$
              Why is $expleft[ipi n(H_{n-1}-1)right]=exp(ipi n)exp(H_{n-1}-1)$?
              $endgroup$
              – greelious
              Jan 17 at 3:29










            • $begingroup$
              @greelious because I messed up... Thanks! :)
              $endgroup$
              – clathratus
              Jan 17 at 3:30










            • $begingroup$
              I think it is still not correct: $expleft[ipi n(H_{n-1}-1)right]=exp(i pi n H_{n-1})exp(-i pi n)=exp(i pi n H_{n-1})(-1)^n$
              $endgroup$
              – greelious
              Jan 17 at 3:37












            • $begingroup$
              @greelious okay look now
              $endgroup$
              – clathratus
              Jan 17 at 3:41










            • $begingroup$
              Looks good now.
              $endgroup$
              – greelious
              Jan 17 at 3:45














            4












            4








            4





            $begingroup$

            $$begin{align}
            f(n)=&prod_{m=2}^{n-1}expfrac{ipi n}{m}\
            &=expleft[sum_{m=2}^{n-1}frac{ipi n}{m}right]\
            &=expleft[ipi nsum_{m=2}^{n-1}frac{1}{m}right]\
            end{align}$$
            Recalling the definition of the harmonic numbers:
            $$H_n=sum_{m=1}^nfrac1m$$
            We have that
            $$f(n)=expleft[ipi n(H_{n-1}-1)right]$$
            Then using $e^{itheta}=costheta+isintheta$,
            $$f(n)=cosleft[pi n(H_{n-1}-1)right]+isinleft[pi n(H_{n-1}-1)right]$$
            So
            $$text{Re}f(n)=cosleft[pi n(H_{n-1}-1)right]$$






            share|cite|improve this answer











            $endgroup$



            $$begin{align}
            f(n)=&prod_{m=2}^{n-1}expfrac{ipi n}{m}\
            &=expleft[sum_{m=2}^{n-1}frac{ipi n}{m}right]\
            &=expleft[ipi nsum_{m=2}^{n-1}frac{1}{m}right]\
            end{align}$$
            Recalling the definition of the harmonic numbers:
            $$H_n=sum_{m=1}^nfrac1m$$
            We have that
            $$f(n)=expleft[ipi n(H_{n-1}-1)right]$$
            Then using $e^{itheta}=costheta+isintheta$,
            $$f(n)=cosleft[pi n(H_{n-1}-1)right]+isinleft[pi n(H_{n-1}-1)right]$$
            So
            $$text{Re}f(n)=cosleft[pi n(H_{n-1}-1)right]$$







            share|cite|improve this answer














            share|cite|improve this answer



            share|cite|improve this answer








            edited Jan 17 at 3:41

























            answered Jan 17 at 3:21









            clathratusclathratus

            3,678332




            3,678332








            • 1




              $begingroup$
              Why is $expleft[ipi n(H_{n-1}-1)right]=exp(ipi n)exp(H_{n-1}-1)$?
              $endgroup$
              – greelious
              Jan 17 at 3:29










            • $begingroup$
              @greelious because I messed up... Thanks! :)
              $endgroup$
              – clathratus
              Jan 17 at 3:30










            • $begingroup$
              I think it is still not correct: $expleft[ipi n(H_{n-1}-1)right]=exp(i pi n H_{n-1})exp(-i pi n)=exp(i pi n H_{n-1})(-1)^n$
              $endgroup$
              – greelious
              Jan 17 at 3:37












            • $begingroup$
              @greelious okay look now
              $endgroup$
              – clathratus
              Jan 17 at 3:41










            • $begingroup$
              Looks good now.
              $endgroup$
              – greelious
              Jan 17 at 3:45














            • 1




              $begingroup$
              Why is $expleft[ipi n(H_{n-1}-1)right]=exp(ipi n)exp(H_{n-1}-1)$?
              $endgroup$
              – greelious
              Jan 17 at 3:29










            • $begingroup$
              @greelious because I messed up... Thanks! :)
              $endgroup$
              – clathratus
              Jan 17 at 3:30










            • $begingroup$
              I think it is still not correct: $expleft[ipi n(H_{n-1}-1)right]=exp(i pi n H_{n-1})exp(-i pi n)=exp(i pi n H_{n-1})(-1)^n$
              $endgroup$
              – greelious
              Jan 17 at 3:37












            • $begingroup$
              @greelious okay look now
              $endgroup$
              – clathratus
              Jan 17 at 3:41










            • $begingroup$
              Looks good now.
              $endgroup$
              – greelious
              Jan 17 at 3:45








            1




            1




            $begingroup$
            Why is $expleft[ipi n(H_{n-1}-1)right]=exp(ipi n)exp(H_{n-1}-1)$?
            $endgroup$
            – greelious
            Jan 17 at 3:29




            $begingroup$
            Why is $expleft[ipi n(H_{n-1}-1)right]=exp(ipi n)exp(H_{n-1}-1)$?
            $endgroup$
            – greelious
            Jan 17 at 3:29












            $begingroup$
            @greelious because I messed up... Thanks! :)
            $endgroup$
            – clathratus
            Jan 17 at 3:30




            $begingroup$
            @greelious because I messed up... Thanks! :)
            $endgroup$
            – clathratus
            Jan 17 at 3:30












            $begingroup$
            I think it is still not correct: $expleft[ipi n(H_{n-1}-1)right]=exp(i pi n H_{n-1})exp(-i pi n)=exp(i pi n H_{n-1})(-1)^n$
            $endgroup$
            – greelious
            Jan 17 at 3:37






            $begingroup$
            I think it is still not correct: $expleft[ipi n(H_{n-1}-1)right]=exp(i pi n H_{n-1})exp(-i pi n)=exp(i pi n H_{n-1})(-1)^n$
            $endgroup$
            – greelious
            Jan 17 at 3:37














            $begingroup$
            @greelious okay look now
            $endgroup$
            – clathratus
            Jan 17 at 3:41




            $begingroup$
            @greelious okay look now
            $endgroup$
            – clathratus
            Jan 17 at 3:41












            $begingroup$
            Looks good now.
            $endgroup$
            – greelious
            Jan 17 at 3:45




            $begingroup$
            Looks good now.
            $endgroup$
            – greelious
            Jan 17 at 3:45










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            cytinus is a new contributor. Be nice, and check out our Code of Conduct.













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            cytinus is a new contributor. Be nice, and check out our Code of Conduct.
















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