# §6.2 Definitions and Interrelations

## §6.2(i) Exponential and Logarithmic Integrals

The principal value of the exponential integral $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ is defined by

 6.2.1 $\mathop{E_{1}\/}\nolimits\!\left(z\right)=\int_{z}^{\infty}\frac{e^{-t}}{t}% \mathrm{d}t,$ $z\neq 0$, Defines: $\mathop{E_{1}\/}\nolimits\!\left(\NVar{z}\right)$: exponential integral Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{e}$: base of exponential function, $\int$: integral and $z$: complex variable A&S Ref: 5.1.1 Referenced by: §6.2(ii), §6.7(iii) Permalink: http://dlmf.nist.gov/6.2.E1 Encodings: TeX, pMML, png See also: Annotations for 6.2(i)

where the path does not cross the negative real axis or pass through the origin. As in the case of the logarithm (§4.2(i)) there is a cut along the interval $(-\infty,0]$ and the principal value is two-valued on $(-\infty,0)$.

Unless indicated otherwise, it is assumed throughout the DLMF that $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ assumes its principal value. This is also true of the functions $\mathop{\mathrm{Ci}\/}\nolimits(z)$ and $\mathop{\mathrm{Chi}\/}\nolimits\!\left(z\right)$ defined in §6.2(ii).

 6.2.2 $\mathop{E_{1}\/}\nolimits\!\left(z\right)=e^{-z}\int_{0}^{\infty}\frac{e^{-t}}% {t+z}\mathrm{d}t,$ $|\mathop{\mathrm{ph}\/}\nolimits z|<\pi$.
 6.2.3 $\mathop{\mathrm{Ein}\/}\nolimits\!\left(z\right)=\int_{0}^{z}\frac{1-e^{-t}}{t% }\mathrm{d}t.$ Defines: $\mathop{\mathrm{Ein}\/}\nolimits\!\left(\NVar{z}\right)$: complementary exponential integral Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{e}$: base of exponential function, $\int$: integral and $z$: complex variable Permalink: http://dlmf.nist.gov/6.2.E3 Encodings: TeX, pMML, png See also: Annotations for 6.2(i)

$\mathop{\mathrm{Ein}\/}\nolimits\!\left(z\right)$ is sometimes called the complementary exponential integral. It is entire.

 6.2.4 $\mathop{E_{1}\/}\nolimits\!\left(z\right)=\mathop{\mathrm{Ein}\/}\nolimits\!% \left(z\right)-\mathop{\ln\/}\nolimits z-\gamma.$

In the next three equations $x>0$.

 6.2.5 $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)=-\,\pvint_{-x}^{\infty}\frac{e% ^{-t}}{t}\mathrm{d}t=\pvint_{-\infty}^{x}\frac{e^{t}}{t}\mathrm{d}t,$
 6.2.6 $\mathop{\mathrm{Ei}\/}\nolimits\!\left(-x\right)=-\int_{x}^{\infty}\frac{e^{-t% }}{t}\mathrm{d}t=-\mathop{E_{1}\/}\nolimits\!\left(x\right),$
 6.2.7 $\mathop{\mathrm{Ei}\/}\nolimits\!\left(\pm x\right)=-\mathop{\mathrm{Ein}\/}% \nolimits\!\left(\mp x\right)+\mathop{\ln\/}\nolimits x+\gamma.$

($\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)$ is undefined when $x=0$, or when $x$ is not real.)

The logarithmic integral is defined by

 6.2.8 $\mathop{\mathrm{li}\/}\nolimits\!\left(x\right)=\pvint_{0}^{x}\frac{\mathrm{d}% t}{\mathop{\ln\/}\nolimits t}=\mathop{\mathrm{Ei}\/}\nolimits\!\left(\mathop{% \ln\/}\nolimits x\right),$ $x>1$. Defines: $\mathop{\mathrm{li}\/}\nolimits\!\left(\NVar{x}\right)$: logarithmic integral Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\mathop{\mathrm{Ei}\/}\nolimits\!\left(\NVar{x}\right)$: exponential integral, $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function, $\pvint_{\NVar{a}}^{\NVar{b}}$: Cauchy principal value and $x$: real variable A&S Ref: 5.1.3 Referenced by: §27.12, §6.12(i) Permalink: http://dlmf.nist.gov/6.2.E8 Encodings: TeX, pMML, png See also: Annotations for 6.2(i)

The generalized exponential integral $\mathop{E_{p}\/}\nolimits\!\left(z\right)$, $p\in\mathbb{C}$, is treated in Chapter 8.

## §6.2(ii) Sine and Cosine Integrals

 6.2.9 $\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)=\int_{0}^{z}\frac{\mathop{\sin% \/}\nolimits t}{t}\mathrm{d}t.$ Defines: $\mathop{\mathrm{Si}\/}\nolimits\!\left(\NVar{z}\right)$: sine integral Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\mathop{\sin\/}\nolimits\NVar{z}$: sine function and $z$: complex variable A&S Ref: 5.2.1 Referenced by: §10.15 Permalink: http://dlmf.nist.gov/6.2.E9 Encodings: TeX, pMML, png See also: Annotations for 6.2(ii)

$\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)$ is an odd entire function.

 6.2.10 $\mathop{\mathrm{si}\/}\nolimits\!\left(z\right)=-\int_{z}^{\infty}\frac{% \mathop{\sin\/}\nolimits t}{t}\mathrm{d}t=\mathop{\mathrm{Si}\/}\nolimits\!% \left(z\right)-\tfrac{1}{2}\pi.$ Defines: $\mathop{\mathrm{si}\/}\nolimits\!\left(\NVar{z}\right)$: sine integral Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\mathop{\sin\/}\nolimits\NVar{z}$: sine function, $\mathop{\mathrm{Si}\/}\nolimits\!\left(\NVar{z}\right)$: sine integral and $z$: complex variable A&S Ref: 5.2.5 (in modified form) 5.2.26 (in modified form) Referenced by: §6.5 Permalink: http://dlmf.nist.gov/6.2.E10 Encodings: TeX, pMML, png See also: Annotations for 6.2(ii)
 6.2.11 $\mathop{\mathrm{Ci}\/}\nolimits(z)=-\int_{z}^{\infty}\frac{\mathop{\cos\/}% \nolimits t}{t}\mathrm{d}t,$ Defines: $\mathop{\mathrm{Ci}\/}\nolimits\!\left(\NVar{z}\right)$: cosine integral Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $z$: complex variable A&S Ref: 5.2.27 Referenced by: §10.15 Permalink: http://dlmf.nist.gov/6.2.E11 Encodings: TeX, pMML, png See also: Annotations for 6.2(ii)

where the path does not cross the negative real axis or pass through the origin. This is the principal value; compare (6.2.1).

 6.2.12 $\mathop{\mathrm{Cin}\/}\nolimits\!\left(z\right)=\int_{0}^{z}\frac{1-\mathop{% \cos\/}\nolimits t}{t}\mathrm{d}t.$ Defines: $\mathop{\mathrm{Cin}\/}\nolimits\!\left(\NVar{z}\right)$: cosine integral Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral and $z$: complex variable A&S Ref: 5.2.2 Permalink: http://dlmf.nist.gov/6.2.E12 Encodings: TeX, pMML, png See also: Annotations for 6.2(ii)

$\mathop{\mathrm{Cin}\/}\nolimits\!\left(z\right)$ is an even entire function.

 6.2.13 $\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right)=-\mathop{\mathrm{Cin}\/}% \nolimits\!\left(z\right)+\mathop{\ln\/}\nolimits z+\gamma.$ Symbols: $\gamma$: Euler’s constant, $\mathop{\mathrm{Ci}\/}\nolimits\!\left(\NVar{z}\right)$: cosine integral, $\mathop{\mathrm{Cin}\/}\nolimits\!\left(\NVar{z}\right)$: cosine integral, $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function and $z$: complex variable A&S Ref: 5.2.2 (in modified form) Referenced by: §6.4 Permalink: http://dlmf.nist.gov/6.2.E13 Encodings: TeX, pMML, png See also: Annotations for 6.2(ii)

### Values at Infinity

 6.2.14 $\displaystyle\lim_{x\to\infty}\mathop{\mathrm{Si}\/}\nolimits\!\left(x\right)$ $\displaystyle=\tfrac{1}{2}\pi,$ $\displaystyle\lim_{x\to\infty}\mathop{\mathrm{Ci}\/}\nolimits\!\left(x\right)$ $\displaystyle=0.$ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathop{\mathrm{Ci}\/}\nolimits\!\left(\NVar{z}\right)$: cosine integral, $\mathop{\mathrm{Si}\/}\nolimits\!\left(\NVar{z}\right)$: sine integral and $x$: real variable A&S Ref: 5.2.25 (extended form of) Referenced by: §6.16(i) Permalink: http://dlmf.nist.gov/6.2.E14 Encodings: TeX, TeX, pMML, pMML, png, png See also: Annotations for 6.2(ii)

### Hyperbolic Analogs of the Sine and Cosine Integrals

 6.2.15 $\displaystyle\mathop{\mathrm{Shi}\/}\nolimits\!\left(z\right)$ $\displaystyle=\int_{0}^{z}\frac{\mathop{\sinh\/}\nolimits t}{t}\mathrm{d}t,$ Defines: $\mathop{\mathrm{Shi}\/}\nolimits\!\left(\NVar{z}\right)$: hyperbolic sine integral Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\mathop{\sinh\/}\nolimits\NVar{z}$: hyperbolic sine function, $\int$: integral and $z$: complex variable A&S Ref: 5.2.3 Permalink: http://dlmf.nist.gov/6.2.E15 Encodings: TeX, pMML, png See also: Annotations for 6.2(ii) 6.2.16 $\displaystyle\mathop{\mathrm{Chi}\/}\nolimits\!\left(z\right)$ $\displaystyle=\gamma+\mathop{\ln\/}\nolimits z+\int_{0}^{z}\frac{\mathop{\cosh% \/}\nolimits t-1}{t}\mathrm{d}t.$ Defines: $\mathop{\mathrm{Chi}\/}\nolimits\!\left(\NVar{z}\right)$: hyperbolic cosine integral Symbols: $\gamma$: Euler’s constant, $\mathrm{d}\NVar{x}$: differential of $x$, $\mathop{\cosh\/}\nolimits\NVar{z}$: hyperbolic cosine function, $\int$: integral, $\mathop{\ln\/}\nolimits\NVar{z}$: principal branch of logarithm function and $z$: complex variable A&S Ref: 5.2.4 Referenced by: §6.4 Permalink: http://dlmf.nist.gov/6.2.E16 Encodings: TeX, pMML, png See also: Annotations for 6.2(ii)

## §6.2(iii) Auxiliary Functions

 6.2.17 $\displaystyle\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ $\displaystyle=\phantom{+}\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right)% \mathop{\sin\/}\nolimits z-\mathop{\mathrm{si}\/}\nolimits\!\left(z\right)% \mathop{\cos\/}\nolimits z,$ Defines: $\mathop{\mathrm{f}\/}\nolimits\!\left(\NVar{z}\right)$: auxiliary function for sine and cosine integrals Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathop{\mathrm{Ci}\/}\nolimits\!\left(\NVar{z}\right)$: cosine integral, $\mathop{\mathrm{si}\/}\nolimits\!\left(\NVar{z}\right)$: sine integral, $\mathop{\sin\/}\nolimits\NVar{z}$: sine function and $z$: complex variable A&S Ref: 5.2.6 Referenced by: §6.4, §6.5 Permalink: http://dlmf.nist.gov/6.2.E17 Encodings: TeX, pMML, png See also: Annotations for 6.2(iii) 6.2.18 $\displaystyle\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ $\displaystyle=-\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right)\mathop{\cos\/}% \nolimits z-\mathop{\mathrm{si}\/}\nolimits\!\left(z\right)\mathop{\sin\/}% \nolimits z.$ Defines: $\mathop{\mathrm{g}\/}\nolimits\!\left(\NVar{z}\right)$: auxiliary function for sine and cosine integrals Symbols: $\mathop{\cos\/}\nolimits\NVar{z}$: cosine function, $\mathop{\mathrm{Ci}\/}\nolimits\!\left(\NVar{z}\right)$: cosine integral, $\mathop{\mathrm{si}\/}\nolimits\!\left(\NVar{z}\right)$: sine integral, $\mathop{\sin\/}\nolimits\NVar{z}$: sine function and $z$: complex variable A&S Ref: 5.2.7 Referenced by: §6.4, §6.5 Permalink: http://dlmf.nist.gov/6.2.E18 Encodings: TeX, pMML, png See also: Annotations for 6.2(iii) 6.2.19 $\displaystyle\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)$ $\displaystyle=\tfrac{1}{2}\pi-\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)% \mathop{\cos\/}\nolimits z-\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)% \mathop{\sin\/}\nolimits z,$ 6.2.20 $\displaystyle\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right)$ $\displaystyle=\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)\mathop{\sin\/}% \nolimits z-\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)\mathop{\cos\/}% \nolimits z.$
 6.2.21 $\displaystyle\frac{\mathrm{d}\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)}{% \mathrm{d}z}$ $\displaystyle=-\mathop{\mathrm{g}\/}\nolimits\!\left(z\right),$ $\displaystyle\frac{\mathrm{d}\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)}{% \mathrm{d}z}$ $\displaystyle=\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)-\frac{1}{z}.$