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6 Exponential, Logarithmic, Sine, and Cosine IntegralsProperties

§6.5 Further Interrelations

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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,
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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)),
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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:
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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),
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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:
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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.
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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:
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