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Ei -- the exponential integral function

Introduction

Ei(x) represents the exponential integral int(exp(-x*t)/t, t=1..infinity).

Call(s)

Ei(x)

Parameters

x - an arithmetical expression

Returns

an arithmetical expression.

Overloadable:

x

Side Effects

When called with a floating point argument, the function is sensitive to the environment variable DIGITS which determines the numerical working precision.

Related Functions

Ci, exp, igamma, int, Si

Details

Example 1

We demonstrate some calls with exact and symbolic input data:

>> Ei(1), Ei(sqrt(2)), Ei(x + 1), Ei(infinity), Ei(-infinity)
                            1/2
                 Ei(1), Ei(2   ), Ei(x + 1), 0, -infinity

Floating point values are computed for floating point arguments:

>> Ei(-1000.0), Ei(1.0), Ei(12.3), Ei(2.0 + 10.0*I)
      - 1.972045137e431 - 3.141592654 I, 0.2193839344,
      
         0.0000003439533949, 0.003675663008 + 0.01234609005 I

Example 2

Ei is singular at the origin:

>> Ei(0)
      Error: singularity [Ei]

The negative real axis is a branch cut of Ei. A jump of height I*2*PI occurs when crossing this cut:

>> Ei(-1.0), Ei(-1.0 + 10^(-10)*I), Ei(-1.0 - 10^(-10)*I)
      - 1.895117816 - 3.141592654 I, - 1.895117816 - 3.141592653 I,
      
         - 1.895117816 + 3.141592653 I

Example 3

The functions diff, float, limit, and series handle expressions involving Ei:

>> diff(Ei(x), x, x, x), float(ln(3 + Ei(sqrt(PI))))
                exp(-x)   2 exp(-x)   2 exp(-x)
              - ------- - --------- - ---------, 1.120796995
                   x          2           3
                             x           x
>> limit(Ei(2*x^2/(1+x)), x = infinity)
                                     0
>> series(Ei(x), x = 0, 3), 
   series(Ei(x), x = infinity, 3),
   series(Ei(x), x = -infinity, 3)
                               2
                              x       3
      - (ln(x) + EULER) + x - -- + O(x ),
                              4
      
         exp(-x)   exp(-x)    / exp(-x) \
         ------- - ------- + O| ------- |,
            x         2       |    3    |
                     x        \   x     /
      
         exp(-x)   exp(-x)    /   exp(-x) \
         ------- - ------- + O| - ------- |
            x         2       |      3    |
                     x        \     x     /

Background

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