§6.5 Further Interrelations

Notes:: For (6.5.1) and (6.5.2) see Olver (1997b, p. 41). (6.5.3) and (6.5.4) follow from (6.6.1), (6.6.2), (6.6.5), and (6.6.6). For (6.5.5) and (6.5.6) see Olver (1997b, p. 42). (6.5.7) follows from (6.2.10), (6.2.17), (6.2.18), (6.5.5), and (6.5.6).
Keywords:: cosine integrals, exponential integrals, sine integrals
Permalink:: http://dlmf.nist.gov/6.5

When $x>0$ ,

6.5.1 $\mathop{E_{1}\/}\nolimits\!\left(-x\pm i0\right)=-\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)\mp i\pi,$

Symbols:: $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral, $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)$ : exponential integral and $x$ : real variable
A&S Ref:: 5.1.7
Referenced by:: §6.5
Permalink:: http://dlmf.nist.gov/6.5.E1
Encodings:: TeX, pMML, png

6.5.2 $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)=-\tfrac{1}{2}(\mathop{E_{1}\/}\nolimits\!\left(-x+i0\right)+\mathop{E_{1}\/}\nolimits\!\left(-x-i0\right)),$

Symbols:: $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral, $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)$ : exponential integral and $x$ : real variable
A&S Ref:: 5.1.7
Referenced by:: §6.5
Permalink:: http://dlmf.nist.gov/6.5.E2
Encodings:: TeX, pMML, png

6.5.3 $\tfrac{1}{2}(\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)+\mathop{E_{1}\/}\nolimits\!\left(x\right))=\mathop{\mathrm{Shi}\/}\nolimits\!\left(x\right)=-i\mathop{\mathrm{Si}\/}\nolimits\!\left(ix\right),$

Symbols:: $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral, $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)$ : exponential integral, $\mathop{\mathrm{Shi}\/}\nolimits\!\left(z\right)$ : hyperbolic sine integral, $\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)$ : sine integral and $x$ : real variable
A&S Ref:: 5.2.22 (in modified form)
Referenced by:: §6.5
Permalink:: http://dlmf.nist.gov/6.5.E3
Encodings:: TeX, pMML, png

6.5.4 $\tfrac{1}{2}(\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)-\mathop{E_{1}\/}\nolimits\!\left(x\right))=\mathop{\mathrm{Chi}\/}\nolimits\!\left(x\right)=\mathop{\mathrm{Ci}\/}\nolimits\!\left(ix\right)-\tfrac{1}{2}\pi i.$

Symbols:: $\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right)$ : cosine integral, $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral, $\mathop{\mathrm{Ei}\/}\nolimits\!\left(x\right)$ : exponential integral, $\mathop{\mathrm{Chi}\/}\nolimits\!\left(z\right)$ : hyperbolic cosine integral and $x$ : real variable
A&S Ref:: 5.2.24 (in modified form)
Referenced by:: §6.5
Permalink:: http://dlmf.nist.gov/6.5.E4
Encodings:: TeX, pMML, png

When $|\mathop{\mathrm{ph}\/}\nolimits z|<\frac{1}{2}\pi$ ,

6.5.5 $\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)=\tfrac{1}{2}i(\mathop{E_{1}\/}\nolimits\!\left(-iz\right)-\mathop{E_{1}\/}\nolimits\!\left(iz\right))+\tfrac{1}{2}\pi,$

Symbols:: $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral, $\mathop{\mathrm{Si}\/}\nolimits\!\left(z\right)$ : sine integral and $z$ : complex variable
A&S Ref:: 5.2.21
Referenced by:: §6.12(ii), §6.5
Permalink:: http://dlmf.nist.gov/6.5.E5
Encodings:: TeX, pMML, png

6.5.6 $\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right)=-\tfrac{1}{2}(\mathop{E_{1}\/}\nolimits\!\left(iz\right)+\mathop{E_{1}\/}\nolimits\!\left(-iz\right)),$

Symbols:: $\mathop{\mathrm{Ci}\/}\nolimits\!\left(z\right)$ : cosine integral, $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral and $z$ : complex variable
A&S Ref:: 5.2.23
Referenced by:: §6.12(ii), §6.5
Permalink:: http://dlmf.nist.gov/6.5.E6
Encodings:: TeX, pMML, png

6.5.7 $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)\pm i\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)=\mathop{E_{1}\/}\nolimits\!\left(\mp iz\right)e^{{\mp iz}}.$

Symbols:: $e$ : base of exponential function, $\mathop{E_{1}\/}\nolimits\!\left(z\right)$ : exponential integral, $\mathop{\mathrm{f}\/}\nolimits\!\left(z\right)$ : auxiliary function for sine and cosine integrals, $\mathop{\mathrm{g}\/}\nolimits\!\left(z\right)$ : auxiliary function for sine and cosine integrals and $z$ : complex variable
Referenced by:: §6.11, §6.5, §6.7(iii)
Permalink:: http://dlmf.nist.gov/6.5.E7
Encodings:: TeX, pMML, png