15.5 Exponential Integrals

The Exponential Integral and related functions are defined in Abramowitz and Stegun, Handbook of Mathematical Functions, A&S Chapter 5.

Function: expintegral_e1 (z) ¶

The Exponential Integral E1(z) defined as

$$E_1(z)={\int }_z^{\mathrm{\infty }}\frac{e^{-t}}{t}dt$$

with $|\arg z|<\pi$ . (A&S eqn 5.1.1) and (DLMF 6.2E2)

Categories: Exponential Integrals · Special functions ·

Function: expintegral_ei (x) ¶

The Exponential Integral Ei(x) defined as

$$Ei(x)=--{\int }_{-x}^{\mathrm{\infty }}\frac{e^{-t}}{t}dt=-{\int }_{-\mathrm{\infty }}^x\frac{e^t}{t}dt$$

with $x$ real and $x>0$. (A&S eqn 5.1.2) and (DLMF 6.2E5)

Categories: Exponential Integrals · Special functions ·

Function: expintegral_li (x) ¶

The Exponential Integral li(x) defined as

$$li(x)=-{\int }_0^x\frac{dt}{\ln t}$$

with $x$ real and $x>1$. (A&S eqn 5.1.3) and (DLMF 6.2E8)

Categories: Exponential Integrals · Special functions ·

Function: expintegral_e (n,z) ¶

The Exponential Integral En(z) (A&S eqn 5.1.4) defined as

$$E_n(z)={\int }_1^{\mathrm{\infty }}\frac{e^{-zt}}{t^n}dt$$

with $\mathrm{R}\mathrm{e}(z)>1$ and $n$ a non-negative integer.

Categories: Exponential Integrals · Special functions ·

Function: expintegral_si (z) ¶

The Exponential Integral Si(z) (A&S eqn 5.2.1) defined as

$$\mathrm{S}\mathrm{i}(z)={\int }_0^z\frac{\sin t}{t}dt$$

Categories: Exponential Integrals · Special functions ·

Function: expintegral_ci (z) ¶

The Exponential Integral Ci(z) (A&S eqn 5.2.2) defined as

$$\mathrm{C}\mathrm{i}(z)=\gamma +\log z+{\int }_0^z\frac{\cos t-1}{t}dt$$

with $|\arg z|<\pi$ .

Categories: Exponential Integrals · Special functions ·

Function: expintegral_shi (z) ¶

The Exponential Integral Shi(z) (A&S eqn 5.2.3) defined as

$$\mathrm{S}\mathrm{h}\mathrm{i}(z)={\int }_0^z\frac{\sinh t}{t}dt$$

Categories: Exponential Integrals · Special functions ·

Function: expintegral_chi (z) ¶

The Exponential Integral Chi(z) (A&S eqn 5.2.4) defined as

$$\mathrm{C}\mathrm{h}\mathrm{i}(z)=\gamma +\log z+{\int }_0^z\frac{\cosh t-1}{t}dt$$

with $|\arg z|<\pi$ .

Categories: Exponential Integrals · Special functions ·

Option variable: expintrep ¶

Default value: false

Change the representation of one of the exponential integrals, expintegral_e(m, z), expintegral_e1, or expintegral_ei to an equivalent form if possible.

Possible values for expintrep are false, gamma_incomplete, expintegral_e1, expintegral_ei, expintegral_li, expintegral_trig, or expintegral_hyp.

false means that the representation is not changed. Other values indicate the representation is to be changed to use the function specified where expintegral_trig means expintegral_si, expintegral_ci, and expintegral_hyp means expintegral_shi or expintegral_chi.

Categories: Exponential Integrals ·

Option variable: expintexpand ¶

Default value: false

Expand expintegral_e(n,z) for half integral values in terms of erfc or erf and for positive integers in terms of expintegral_ei.

Categories: Exponential Integrals ·