§6.2 Definitions and Interrelations

§6.2(i) Exponential and Logarithmic Integrals

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

 6.2.1 $E_{1}\left(z\right)=\int_{z}^{\infty}\frac{e^{-t}}{t}\mathrm{d}t,$ $z\neq 0$, ⓘ Defines: $E_{1}\left(\NVar{z}\right)$: 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), 6.2 and 6

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 $E_{1}\left(z\right)$ assumes its principal value. This is also true of the functions $\mathrm{Ci}(z)$ and $\mathrm{Chi}\left(z\right)$ defined in §6.2(ii).

 6.2.2 $E_{1}\left(z\right)=e^{-z}\int_{0}^{\infty}\frac{e^{-t}}{t+z}\mathrm{d}t,$ $|\operatorname{ph}z|<\pi$.
 6.2.3 $\mathrm{Ein}\left(z\right)=\int_{0}^{z}\frac{1-e^{-t}}{t}\mathrm{d}t.$ ⓘ Defines: $\mathrm{Ein}\left(\NVar{z}\right)$: 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), 6.2 and 6

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

 6.2.4 $E_{1}\left(z\right)=\mathrm{Ein}\left(z\right)-\ln z-\gamma.$

In the next three equations $x>0$.

 6.2.5 $\mathrm{Ei}\left(x\right)=-\,\pvint_{-x}^{\infty}\frac{e^{-t}}{t}\mathrm{d}t=% \pvint_{-\infty}^{x}\frac{e^{t}}{t}\mathrm{d}t,$
 6.2.6 $\mathrm{Ei}\left(-x\right)=-\int_{x}^{\infty}\frac{e^{-t}}{t}\mathrm{d}t=-E_{1% }\left(x\right),$
 6.2.7 $\mathrm{Ei}\left(\pm x\right)=-\mathrm{Ein}\left(\mp x\right)+\ln x+\gamma.$ ⓘ Symbols: $\gamma$: Euler’s constant, $\mathrm{Ein}\left(\NVar{z}\right)$: complementary exponential integral, $\mathrm{Ei}\left(\NVar{x}\right)$: exponential integral, $\ln\NVar{z}$: principal branch of logarithm function and $x$: real variable A&S Ref: 5.1.40 (in modified form) Permalink: http://dlmf.nist.gov/6.2.E7 Encodings: TeX, pMML, png See also: Annotations for 6.2(i), 6.2 and 6

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

The logarithmic integral is defined by

 6.2.8 $\mathrm{li}\left(x\right)=\pvint_{0}^{x}\frac{\mathrm{d}t}{\ln t}=\mathrm{Ei}% \left(\ln x\right),$ $x>1$. ⓘ Defines: $\mathrm{li}\left(\NVar{x}\right)$: logarithmic integral Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\mathrm{Ei}\left(\NVar{x}\right)$: exponential integral, $\ln\NVar{z}$: 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), 6.2 and 6

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

§6.2(ii) Sine and Cosine Integrals

 6.2.9 $\mathrm{Si}\left(z\right)=\int_{0}^{z}\frac{\sin t}{t}\mathrm{d}t.$ ⓘ Defines: $\mathrm{Si}\left(\NVar{z}\right)$: sine integral Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\sin\NVar{z}$: 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), 6.2 and 6

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

 6.2.10 $\mathrm{si}\left(z\right)=-\int_{z}^{\infty}\frac{\sin t}{t}\mathrm{d}t=% \mathrm{Si}\left(z\right)-\tfrac{1}{2}\pi.$ ⓘ Defines: $\mathrm{si}\left(\NVar{z}\right)$: sine integral Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{d}\NVar{x}$: differential of $x$, $\int$: integral, $\sin\NVar{z}$: sine function, $\mathrm{Si}\left(\NVar{z}\right)$: 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 and 6
 6.2.11 $\mathrm{Ci}(z)=-\int_{z}^{\infty}\frac{\cos t}{t}\mathrm{d}t,$ ⓘ Defines: $\mathrm{Ci}\left(\NVar{z}\right)$: cosine integral Symbols: $\cos\NVar{z}$: 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), 6.2 and 6

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 $\mathrm{Cin}\left(z\right)=\int_{0}^{z}\frac{1-\cos t}{t}\mathrm{d}t.$ ⓘ Defines: $\mathrm{Cin}\left(\NVar{z}\right)$: cosine integral Symbols: $\cos\NVar{z}$: 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), 6.2 and 6

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

 6.2.13 $\mathrm{Ci}\left(z\right)=-\mathrm{Cin}\left(z\right)+\ln z+\gamma.$ ⓘ Symbols: $\gamma$: Euler’s constant, $\mathrm{Ci}\left(\NVar{z}\right)$: cosine integral, $\mathrm{Cin}\left(\NVar{z}\right)$: cosine integral, $\ln\NVar{z}$: 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), 6.2 and 6

Values at Infinity

 6.2.14 $\displaystyle\lim_{x\to\infty}\mathrm{Si}\left(x\right)$ $\displaystyle=\tfrac{1}{2}\pi,$ $\displaystyle\lim_{x\to\infty}\mathrm{Ci}\left(x\right)$ $\displaystyle=0.$ ⓘ Symbols: $\pi$: the ratio of the circumference of a circle to its diameter, $\mathrm{Ci}\left(\NVar{z}\right)$: cosine integral, $\mathrm{Si}\left(\NVar{z}\right)$: 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), 6.2(ii), 6.2 and 6

Hyperbolic Analogs of the Sine and Cosine Integrals

 6.2.15 $\displaystyle\mathrm{Shi}\left(z\right)$ $\displaystyle=\int_{0}^{z}\frac{\sinh t}{t}\mathrm{d}t,$ ⓘ Defines: $\mathrm{Shi}\left(\NVar{z}\right)$: hyperbolic sine integral Symbols: $\mathrm{d}\NVar{x}$: differential of $x$, $\sinh\NVar{z}$: 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(ii), 6.2 and 6 6.2.16 $\displaystyle\mathrm{Chi}\left(z\right)$ $\displaystyle=\gamma+\ln z+\int_{0}^{z}\frac{\cosh t-1}{t}\mathrm{d}t.$ ⓘ Defines: $\mathrm{Chi}\left(\NVar{z}\right)$: hyperbolic cosine integral Symbols: $\gamma$: Euler’s constant, $\mathrm{d}\NVar{x}$: differential of $x$, $\cosh\NVar{z}$: hyperbolic cosine function, $\int$: integral, $\ln\NVar{z}$: 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(ii), 6.2 and 6

§6.2(iii) Auxiliary Functions

 6.2.17 $\displaystyle\mathrm{f}\left(z\right)$ $\displaystyle=\phantom{+}\mathrm{Ci}\left(z\right)\sin z-\mathrm{si}\left(z% \right)\cos z,$ ⓘ Defines: $\mathrm{f}\left(\NVar{z}\right)$: auxiliary function for sine and cosine integrals Symbols: $\cos\NVar{z}$: cosine function, $\mathrm{Ci}\left(\NVar{z}\right)$: cosine integral, $\mathrm{si}\left(\NVar{z}\right)$: sine integral, $\sin\NVar{z}$: 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 and 6 6.2.18 $\displaystyle\mathrm{g}\left(z\right)$ $\displaystyle=-\mathrm{Ci}\left(z\right)\cos z-\mathrm{si}\left(z\right)\sin z.$ ⓘ Defines: $\mathrm{g}\left(\NVar{z}\right)$: auxiliary function for sine and cosine integrals Symbols: $\cos\NVar{z}$: cosine function, $\mathrm{Ci}\left(\NVar{z}\right)$: cosine integral, $\mathrm{si}\left(\NVar{z}\right)$: sine integral, $\sin\NVar{z}$: 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 and 6 6.2.19 $\displaystyle\mathrm{Si}\left(z\right)$ $\displaystyle=\tfrac{1}{2}\pi-\mathrm{f}\left(z\right)\cos z-\mathrm{g}\left(z% \right)\sin z,$ 6.2.20 $\displaystyle\mathrm{Ci}\left(z\right)$ $\displaystyle=\mathrm{f}\left(z\right)\sin z-\mathrm{g}\left(z\right)\cos z.$
 6.2.21 $\displaystyle\frac{\mathrm{d}\mathrm{f}\left(z\right)}{\mathrm{d}z}$ $\displaystyle=-\mathrm{g}\left(z\right),$ $\displaystyle\frac{\mathrm{d}\mathrm{g}\left(z\right)}{\mathrm{d}z}$ $\displaystyle=\mathrm{f}\left(z\right)-\frac{1}{z}.$