§6.4 Analytic Continuation

Notes:: For (6.4.1) see Olver (1997b, p. 40). (6.4.3) follows from (6.6.2) and (6.6.4). (6.4.4) and (6.4.5) follow from (6.2.13) and (6.2.16). (6.4.6) and (6.4.7) follow from (6.2.17), (6.2.18), and (6.4.4).
Keywords:: analytic continuation, auxiliary functions for sine and cosine integrals, cosine integrals, exponential integrals, principal values
Referenced by:: §6.18(i)
Permalink:: http://dlmf.nist.gov/6.4

Analytic continuation of the principal value of $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ yields a multi-valued function with branch points at $z=0$ and $z=\infty$ . The general value of $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ is given by

6.4.1 $\mathop{E_{1}\/}\nolimits\!\left(z\right)=\mathop{\mathrm{Ein}\/}\nolimits\!\left(z\right)-\mathop{\mathrm{Ln}\/}\nolimits z-\EulerConstant;$

Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{\mathrm{Ein}\/}\nolimits\!\left(z\right)$ : complementary exponential integral, $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral, $\mathop{\mathrm{Ln}\/}\nolimits z$ : general logarithm function and $z$ : complex variable
Referenced by:: §6.4
Permalink:: http://dlmf.nist.gov/6.4.E1
Encodings:: TeX, pMML, png

compare (6.2.4) and (4.2.6). Thus

6.4.2 $\mathop{E_{1}\/}\nolimits\!\left(ze^{{2m\pi i}}\right)=\mathop{E_{1}\/}\nolimits\!\left(z\right)-2m\pi i,$ $m\in\Integer$ ,

Symbols:: $\in$ : element of, $e$ : base of exponential function, $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral, $\Integer$ : set of all integers and $z$ : complex variable
Permalink:: http://dlmf.nist.gov/6.4.E2
Encodings:: TeX, pMML, png

and

6.4.3 $\mathop{E_{1}\/}\nolimits\!\left(ze^{{\pm\pi i}}\right)=\mathop{\mathrm{Ein}\/}\nolimits\!\left(-z\right)-\mathop{\ln\/}\nolimits z-\EulerConstant\mp\pi i,$ $|\mathop{\mathrm{ph}\/}\nolimits z|\leq\pi$ .

Symbols:: $\EulerConstant$ : Euler’s constant, $\mathop{\mathrm{Ein}\/}\nolimits\!\left(z\right)$ : complementary exponential integral, $e$ : base of exponential function, $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral, $\mathop{\ln\/}\nolimits z$ : principal branch of logarithm function, $\mathop{\mathrm{ph}\/}\nolimits$ : phase and $z$ : complex variable
Referenced by:: §6.4
Permalink:: http://dlmf.nist.gov/6.4.E3
Encodings:: TeX, pMML, png

The general values of the other functions are defined in a similar manner, and

6.4.4 $\mathop{\mathrm{Ci}\/}\nolimits\!\left(ze^{{\pm\pi i}}\right)=\pm\pi i+\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right),$

Symbols:: $\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right)$ : cosine integral, $e$ : base of exponential function and $z$ : complex variable
Referenced by:: §6.4
Permalink:: http://dlmf.nist.gov/6.4.E4
Encodings:: TeX, pMML, png

6.4.5 $\mathop{\mathrm{Chi}\/}\nolimits\!\left(ze^{{\pm\pi i}}\right)=\pm\pi i+\mathop{\mathrm{Chi}\/}\nolimits\!\left(z\right),$

Symbols:: $e$ : base of exponential function, $\mathop{\mathrm{Chi}\/}\nolimits\!\left(z\right)$ : hyperbolic cosine integral and $z$ : complex variable
Referenced by:: §6.4
Permalink:: http://dlmf.nist.gov/6.4.E5
Encodings:: TeX, pMML, png

6.4.6 $\mathop{\mathrm{f}\/}\nolimits\!\left(ze^{{\pm\pi i}}\right)=\pi e^{{\mp iz}}-\mathop{\mathrm{f}\/}\nolimits\!\left(z\right),$

Symbols:: $e$ : base of exponential function, $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for sine and cosine integrals and $z$ : complex variable
Referenced by:: §6.12(ii), §6.4
Permalink:: http://dlmf.nist.gov/6.4.E6
Encodings:: TeX, pMML, png

6.4.7 $\mathop{\mathrm{g}\/}\nolimits\!\left(ze^{{\pm\pi i}}\right)=\mp\pi ie^{{\mp iz}}+\mathop{\mathrm{g}\/}\nolimits\!\left(z\right).$

Symbols:: $e$ : base of exponential function, $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : auxiliary function for sine and cosine integrals and $z$ : complex variable
Referenced by:: §6.12(ii), §6.4
Permalink:: http://dlmf.nist.gov/6.4.E7
Encodings:: TeX, pMML, png

Unless indicated otherwise, in the rest of this chapter and elsewhere in the DLMF the functions $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ , $\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right)$ , $\mathop{\mathrm{Chi}\/}\nolimits\!\left(z\right)$ , $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ , and $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ assume their principal values, that is, the branches that are real on the positive real axis and two-valued on the negative real axis.