Kummer's function
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In mathematics, Kummer's function, named after Ernst Kummer, is a complex function related to the polylogarithm. It is defined by : \Lambda_n(z)=\int_0^z \frac{\log^{n-1}|t|}{1+t}\;dt. Its duplication formula is : \Lambda_n(z)+\Lambda_n(-z)= 2^{1-n}\Lambda_n(-z^2) . Compare this to the duplication formula for the polylogarithm: : \operatorname{Li}_n(z)+\operatorname{Li}_n(-z)= 2^{1-n}\operatorname{Li}_n(z^2). An explicit link to the polylogarithm is given by : \operatorname{Li}_n(z)=\operatorname{Li}_n(1)\;\;+\;\; \sum_{k=1}^{n-1} (-1)^{k-1} \;\frac{\log^k |z|} {k!} \;\operatorname{Li}_{n-k} (z) \;\;+\;\; \frac{(-1)^{n-1}}{(n-1)!} \;\left[ \Lambda_n(-1) - \Lambda_n(-z) \right].
In mathematics, Kummer's function, named after Ernst Kummer, is a complex function related to the polylogarithm. It is defined by \Lambda_n(z)=\int_0^z \frac{\log^{n-1}|t|}{1+t}\;dt. Its duplication formula is \Lambda_n(z)+\Lambda_n(-z)= 2^{1-n}\Lambda_n(-z^2) . Compare this to the duplication formula for the polylogarithm: \operatorname{Li}_n(z)+\operatorname{Li}_n(-z)= 2^{1-n}\operatorname{Li}_n(z^2). An explicit link to the polylogarithm is given by \operatorname{Li}_n(z)=\operatorname{Li}_n(1)\;\;+\;\; \sum_{k=1}^{n-1} (-1)^{k-1} \;\frac{\log^k |z|} {k!} \;\operatorname{Li}_{n-k} (z) \;\;+\;\; \frac{(-1)^{n-1}}{(n-1)!} \;\left[ \Lambda_n(-1) - \Lambda_n(-z) \right]. ==References== hu:Kummer-függvény
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