Difference between revisions of "Relationship between logarithmic integral and exponential integral"
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| − | + | ==Theorem== | |
| − | + | The following formula holds: | |
$$\mathrm{li}(x)=\mathrm{Ei}( \log(x)),$$ | $$\mathrm{li}(x)=\mathrm{Ei}( \log(x)),$$ | ||
where $\mathrm{li}$ denotes the [[logarithmic integral]], $\mathrm{Ei}$ denotes the [[exponential integral Ei]], and $\log$ denotes the [[logarithm]]. | where $\mathrm{li}$ denotes the [[logarithmic integral]], $\mathrm{Ei}$ denotes the [[exponential integral Ei]], and $\log$ denotes the [[logarithm]]. | ||
| − | + | ||
| − | < | + | ==Proof== |
| − | + | ||
| − | + | ==References== | |
| + | * {{PaperReference|On certain definite integrals involving the exponential-integral|1881|James Whitbread Lee Glaisher|prev=Logarithmic integral|next=findme}} (<i>note: expresses this relationship as $\mathrm{Ei}(x)=\mathrm{li}(e^x)$</i>) | ||
| + | |||
| + | [[Category:Theorem]] | ||
| + | [[Category:Unproven]] | ||
Latest revision as of 03:34, 17 March 2018
Theorem
The following formula holds: $$\mathrm{li}(x)=\mathrm{Ei}( \log(x)),$$ where $\mathrm{li}$ denotes the logarithmic integral, $\mathrm{Ei}$ denotes the exponential integral Ei, and $\log$ denotes the logarithm.
Proof
References
- James Whitbread Lee Glaisher: On certain definite integrals involving the exponential-integral (1881)... (previous)... (next) (note: expresses this relationship as $\mathrm{Ei}(x)=\mathrm{li}(e^x)$)
