Difference between revisions of "Logarithmic integral"
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(Created page with "The logarithmic integral is $$\mathrm{li}(x) = \displaystyle\int_0^x \dfrac{dt}{\log(t)},$$ where $\log$ denotes the logarithm. It can be shown that $\mathrm{li}(x)=\mathr...") |
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The logarithmic integral is | The logarithmic integral is | ||
| − | $$\mathrm{li}(x) = \displaystyle\int_0^x \dfrac{ | + | $$\mathrm{li}(x) = \displaystyle\int_0^x \dfrac{1}{\log(t)} \mathrm{d}t,$$ |
| − | where $\log$ denotes the [[logarithm]]. | + | where $\log$ denotes the [[logarithm]]. |
| + | |||
| + | <div align="center"> | ||
| + | <gallery> | ||
| + | File:Liplot.png|Graph of $\mathrm{li}$. | ||
| + | File:Complexliplot.png|[[Domain coloring]] of $\mathrm{li}$. | ||
| + | </gallery> | ||
| + | </div> | ||
| + | |||
| + | =Properties= | ||
| + | [[Relationship between logarithmic integral and exponential integral]]<br /> | ||
| + | [[Prime number theorem, logarithmic integral]]<br /> | ||
| + | |||
| + | =See Also= | ||
| + | [[Prime counting function]] <br /> | ||
| + | |||
| + | =References= | ||
| + | * {{PaperReference|On certain definite integrals involving the exponential-integral|1881|James Whitbread Lee Glaisher|prev=Exponential integral Ei series|next=Relationship between logarithmic integral and exponential integral}} | ||
| + | * {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Exponential integral Ei|next=Exponential integral E}}: $5.1.3$ | ||
| + | |||
| + | {{:Logarithm and friends footer}} | ||
| + | {{:*-integral functions footer}} | ||
| + | |||
| + | |||
| + | [[Category:SpecialFunction]] | ||
Latest revision as of 03:33, 17 March 2018
The logarithmic integral is $$\mathrm{li}(x) = \displaystyle\int_0^x \dfrac{1}{\log(t)} \mathrm{d}t,$$ where $\log$ denotes the logarithm.
Graph of $\mathrm{li}$.
Domain coloring of $\mathrm{li}$.
Properties
Relationship between logarithmic integral and exponential integral
Prime number theorem, logarithmic integral
See Also
References
- James Whitbread Lee Glaisher: On certain definite integrals involving the exponential-integral (1881)... (previous)... (next)
- 1964: Milton Abramowitz and Irene A. Stegun: Handbook of mathematical functions ... (previous) ... (next): $5.1.3$
