Difference between revisions of "Exponential integral Ei"

From specialfunctionswiki
Jump to: navigation, search
 
(10 intermediate revisions by the same user not shown)
Line 1: Line 1:
The exponential integral $\mathrm{Ei}$ is defined by
+
The exponential integral $\mathrm{Ei}$ is defined for $x>0$ by
$$\mathrm{Ei}(z) = \int_{-\infty}^x \dfrac{e^t}{t} \mathrm{d}t, \quad \left|\mathrm{arg}(-z) \right|<\pi.$$
+
$$\mathrm{Ei}(x) = \mathrm{PV}\int_{-\infty}^x \dfrac{e^t}{t} \mathrm{d}t,$$
 +
where $\mathrm{PV}$ denotes the [[Cauchy principal value]] and $e^t$ denotes the [[exponential]].
  
  
Line 11: Line 12:
  
 
=Properties=
 
=Properties=
 +
[[Ei(-x)=-Integral from -x to infinity of e^(-t)/t dt]]<br />
 
[[Relationship between logarithmic integral and exponential integral]]<br />
 
[[Relationship between logarithmic integral and exponential integral]]<br />
 
[[Exponential integral Ei series]]<br />
 
[[Exponential integral Ei series]]<br />
 
[[Relationship between exponential integral Ei, cosine integral, and sine integral]]<br />
 
[[Relationship between exponential integral Ei, cosine integral, and sine integral]]<br />
 +
 +
=See Also=
 +
[[Exponential integral E]]
  
 
=References=
 
=References=
[http://dualaud.net/specialfunctionswiki/abramowitz_and_stegun-1.03/page_228.htm Exponential Integral and Related Functions]<br />
+
* {{PaperReference|On certain definite integrals involving the exponential-integral|1881|James Whitbread Lee Glaisher|next=Ei(-x)=-Integral from x to infinity of e^(-t)/t dt}}
[http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN600494829_0018%7CLOG_0048 On certain definite integrals involving the exponential-integral - J.W.L. Glaisher]
+
* {{BookReference|Handbook of mathematical functions|1964|Milton Abramowitz|author2=Irene A. Stegun|prev=Ei(-x)=-Integral from -x to infinity of e^(-t)/t dt|next=Logarithmic integral}}: $5.1.2$
 
 
 
{{:*-integral functions footer}}
 
{{:*-integral functions footer}}
  
 
[[Category:SpecialFunction]]
 
[[Category:SpecialFunction]]

Latest revision as of 00:48, 24 March 2018

The exponential integral $\mathrm{Ei}$ is defined for $x>0$ by $$\mathrm{Ei}(x) = \mathrm{PV}\int_{-\infty}^x \dfrac{e^t}{t} \mathrm{d}t,$$ where $\mathrm{PV}$ denotes the Cauchy principal value and $e^t$ denotes the exponential.


Properties

Ei(-x)=-Integral from -x to infinity of e^(-t)/t dt
Relationship between logarithmic integral and exponential integral
Exponential integral Ei series
Relationship between exponential integral Ei, cosine integral, and sine integral

See Also

Exponential integral E

References

$\ast$-integral functions