Are there any non-trivial special values of $operatorname{Li}_4(z)$?
$begingroup$
Denote $operatorname{Li}_4(z)$ the analytic continuation of $sum_{n=1}^inftyfrac{z^n}{n^4}$. $z$ is a algebraic number with $|z|ne 0,1$. Does $Reoperatorname{Li}_4(z)$ or $Imoperatorname{Li}_4(z)$ have closed-form with some $z$?
The following is something I've found.
$$operatorname{Li}_4(0)=0$$
$$operatorname{Li}_4(1)=zeta(4)$$
$$operatorname{Li}_4(-1)=-eta(4)$$
$$operatorname{Li}_4(i)=-frac{7 pi ^4}{11520}+frac{i psi ^{(3)}left(frac{1}{4}right)}{1536}-frac{i psi ^{(3)}left(frac{3}{4}right)}{1536}$$
$$operatorname{Li}_4(-i)=-frac{7 pi ^4}{11520}-frac{i psi ^{(3)}left(frac{1}{4}right)}{1536}+frac{i psi ^{(3)}left(frac{3}{4}right)}{1536}$$
$$Reoperatorname{Li}_4(e^{ix})=-frac{x^4}{48}+frac{pi x^3}{12}-frac{pi ^2 x^2}{12}+frac{pi ^4}{90}$$
But I failed to give another non-trivial examples. I thought about $Reoperatorname{Li}_4left(frac{1+i}2right)$ and use the same method evaluating $int_0^1 frac{x log ^2(x+1)}{x^2+1} , dx$ ($xmapstofrac{1-x}{1+x}$) to evaluate $$int_0^1 frac{x log ^3(x+1)}{x^2+1} , dx$$ but failed.
calculus closed-form polylogarithm
asked Dec 4 '18 at 8:28
Kemono ChenKemono Chen
2,9651739
$endgroup$
add a comment |
$begingroup$
Denote $operatorname{Li}_4(z)$ the analytic continuation of $sum_{n=1}^inftyfrac{z^n}{n^4}$. $z$ is a algebraic number with $|z|ne 0,1$. Does $Reoperatorname{Li}_4(z)$ or $Imoperatorname{Li}_4(z)$ have closed-form with some $z$?
The following is something I've found.
$$operatorname{Li}_4(0)=0$$
$$operatorname{Li}_4(1)=zeta(4)$$
$$operatorname{Li}_4(-1)=-eta(4)$$
$$operatorname{Li}_4(i)=-frac{7 pi ^4}{11520}+frac{i psi ^{(3)}left(frac{1}{4}right)}{1536}-frac{i psi ^{(3)}left(frac{3}{4}right)}{1536}$$
$$operatorname{Li}_4(-i)=-frac{7 pi ^4}{11520}-frac{i psi ^{(3)}left(frac{1}{4}right)}{1536}+frac{i psi ^{(3)}left(frac{3}{4}right)}{1536}$$
$$Reoperatorname{Li}_4(e^{ix})=-frac{x^4}{48}+frac{pi x^3}{12}-frac{pi ^2 x^2}{12}+frac{pi ^4}{90}$$
But I failed to give another non-trivial examples. I thought about $Reoperatorname{Li}_4left(frac{1+i}2right)$ and use the same method evaluating $int_0^1 frac{x log ^2(x+1)}{x^2+1} , dx$ ($xmapstofrac{1-x}{1+x}$) to evaluate $$int_0^1 frac{x log ^3(x+1)}{x^2+1} , dx$$ but failed.
calculus closed-form polylogarithm
asked Dec 4 '18 at 8:28
Kemono ChenKemono Chen
2,9651739
$endgroup$
add a comment |
$begingroup$
Denote $operatorname{Li}_4(z)$ the analytic continuation of $sum_{n=1}^inftyfrac{z^n}{n^4}$. $z$ is a algebraic number with $|z|ne 0,1$. Does $Reoperatorname{Li}_4(z)$ or $Imoperatorname{Li}_4(z)$ have closed-form with some $z$?
The following is something I've found.
$$operatorname{Li}_4(0)=0$$
$$operatorname{Li}_4(1)=zeta(4)$$
$$operatorname{Li}_4(-1)=-eta(4)$$
$$operatorname{Li}_4(i)=-frac{7 pi ^4}{11520}+frac{i psi ^{(3)}left(frac{1}{4}right)}{1536}-frac{i psi ^{(3)}left(frac{3}{4}right)}{1536}$$
$$operatorname{Li}_4(-i)=-frac{7 pi ^4}{11520}-frac{i psi ^{(3)}left(frac{1}{4}right)}{1536}+frac{i psi ^{(3)}left(frac{3}{4}right)}{1536}$$
$$Reoperatorname{Li}_4(e^{ix})=-frac{x^4}{48}+frac{pi x^3}{12}-frac{pi ^2 x^2}{12}+frac{pi ^4}{90}$$
But I failed to give another non-trivial examples. I thought about $Reoperatorname{Li}_4left(frac{1+i}2right)$ and use the same method evaluating $int_0^1 frac{x log ^2(x+1)}{x^2+1} , dx$ ($xmapstofrac{1-x}{1+x}$) to evaluate $$int_0^1 frac{x log ^3(x+1)}{x^2+1} , dx$$ but failed.
calculus closed-form polylogarithm
asked Dec 4 '18 at 8:28
Kemono ChenKemono Chen
2,9651739
$endgroup$
Denote $operatorname{Li}_4(z)$ the analytic continuation of $sum_{n=1}^inftyfrac{z^n}{n^4}$. $z$ is a algebraic number with $|z|ne 0,1$. Does $Reoperatorname{Li}_4(z)$ or $Imoperatorname{Li}_4(z)$ have closed-form with some $z$?
The following is something I've found.
$$operatorname{Li}_4(0)=0$$
$$operatorname{Li}_4(1)=zeta(4)$$
$$operatorname{Li}_4(-1)=-eta(4)$$
$$operatorname{Li}_4(i)=-frac{7 pi ^4}{11520}+frac{i psi ^{(3)}left(frac{1}{4}right)}{1536}-frac{i psi ^{(3)}left(frac{3}{4}right)}{1536}$$
$$operatorname{Li}_4(-i)=-frac{7 pi ^4}{11520}-frac{i psi ^{(3)}left(frac{1}{4}right)}{1536}+frac{i psi ^{(3)}left(frac{3}{4}right)}{1536}$$
$$Reoperatorname{Li}_4(e^{ix})=-frac{x^4}{48}+frac{pi x^3}{12}-frac{pi ^2 x^2}{12}+frac{pi ^4}{90}$$
But I failed to give another non-trivial examples. I thought about $Reoperatorname{Li}_4left(frac{1+i}2right)$ and use the same method evaluating $int_0^1 frac{x log ^2(x+1)}{x^2+1} , dx$ ($xmapstofrac{1-x}{1+x}$) to evaluate $$int_0^1 frac{x log ^3(x+1)}{x^2+1} , dx$$ but failed.
calculus closed-form polylogarithm
calculus closed-form polylogarithm
asked Dec 4 '18 at 8:28
Kemono ChenKemono Chen
2,9651739
asked Dec 4 '18 at 8:28
Kemono ChenKemono Chen
2,9651739
asked Dec 4 '18 at 8:28
Kemono ChenKemono Chen
2,9651739
asked Dec 4 '18 at 8:28
Kemono ChenKemono Chen
2,9651739
asked Dec 4 '18 at 8:28
Kemono ChenKemono Chen
2,9651739
2,9651739
add a comment |
add a comment |
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