Difference between revisions of "Logarithmic integral"

From specialfunctionswiki
Jump to: navigation, search
Line 6: Line 6:
 
<gallery>
 
<gallery>
 
File:Logarithmicintegral.png|Graph of $\mathrm{li}$ on $[0,6]$.
 
File:Logarithmicintegral.png|Graph of $\mathrm{li}$ on $[0,6]$.
 +
File:Domain coloring of log integral.png|[[Domain coloring]] of [[analytic continuation]] of $\mathrm{li}$.
 
</gallery>
 
</gallery>
 
</div>
 
</div>

Revision as of 19:08, 25 July 2015

The logarithmic integral is $$\mathrm{li}(x) = \displaystyle\int_0^x \dfrac{dt}{\log(t)},$$ where $\log$ denotes the logarithm.

Properties

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

Theorem

The following formula holds: $$\lim_{x \rightarrow \infty} \dfrac{\pi(x)}{\mathrm{li}(x)}=1,$$ where $\pi$ denotes the prime counting function and $\mathrm{li}$ denotes the logarithmic integral.

Proof

References