The exponential integrals are $$\mathrm{Ei}(z) = \int_{-\infty}^x \dfrac{e^t}{t} dt; |\mathrm{arg}(-z)|<\pi,$$ $$E_1(z) = \displaystyle\int_z^{\infty} \dfrac{e^{-t}}{t}dt;|\mathrm{arg \hspace{2pt}}z|<\pi,$$ and $$E_n(z)=\displaystyle\int_1^{\infty} \dfrac{e^{-zt}}{t^n} dt.$$

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

James Whitbread Lee Glaisher: On certain definite integrals involving the exponential-integral (1881)... (previous)... (next) (note: expresses this relationship as $\mathrm{Ei}(x)=\mathrm{li}(e^x)$)

Theorem

The following formula holds for $x>0$: $$\mathrm{Ei}(x) = \gamma + \log x + \displaystyle\sum_{k=1}^{\infty} \dfrac{x^k}{kk!},$$ where $\mathrm{Ei}$ denotes the exponential integral Ei, $\log$ denotes the logarithm, and $\gamma$ denotes the Euler-Mascheroni constant.

Proof

References

James Whitbread Lee Glaisher: On certain definite integrals involving the exponential-integral (1881)... (previous)... (next) (note: expresses the logarithm term as $\frac{1}{4}\log(x^4)$)

Theorem: The exponential integral $E_1$ has series representation $$E_1(z)=-\gamma-\log z - \displaystyle\sum_{k=1}^{\infty} \dfrac{(-1)^kz^k}{kk!}; |\mathrm{arg}(z)|<\pi,$$ where $\gamma$ denotes the Euler-Mascheroni constant.

Proof: █

Theorem (Symmetry): The following symmetry relation holds: $$E_n(\overline{z})=\overline{E_n(z)}.$$

Proof: █

Theorem (Recurrence): The following recurrence holds: $$E_{n+1}(z) = \dfrac{1}{n}[e^{-z}-zE_n(z)];(n=1,2,3,\ldots).$$

Proof: █

Theorem (Continued fraction): The following formula holds: $$E_n(z)=e^{-z} \left( \dfrac{1}{z+} \dfrac{n}{1+} \dfrac{1}{z+} \dfrac{n+1}{1+} \dfrac{2}{z+} \ldots \right); |\mathrm{arg} z|<\pi.$$

Proof: █

Theorem: The following value is known: $$E_n(0)=\dfrac{1}{n-1}; n>1.$$

Proof: █

Theorem: The following closed form expression is known: $$E_0(z)=\dfrac{e^{-z}}{z}.$$

Proof: █

Theorem (Derivative): $$\dfrac{d}{dz} E_n(z) = -E_{n-1}(z); n=1,2,3,\ldots$$

Proof: █

Theorem

The following formula holds: $$E_n(z)=z^{n-1}\Gamma(1-n,z),$$ where $E_n$ denotes the exponential integral E and $\Gamma$ denotes the incomplete gamma function.

Proof

References

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.