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One-argument exponential integral function




ei(x) returns the one-argument exponential integral defined as



Exponential Integral for Floating-Point and Symbolic Numbers

Compute exponential integrals for numeric inputs. Because these numbers are not symbolic objects, you get floating-point results.

s = [ei(-2), ei(-1/2), ei(1), ei(sqrt(2))]
s =
   -0.0489   -0.5598    1.8951    3.0485

Compute exponential integrals for the same numbers converted to symbolic objects. For most symbolic (exact) numbers, ei returns unresolved symbolic calls.

s = [ei(sym(-2)), ei(sym(-1/2)), ei(sym(1)), ei(sqrt(sym(2)))]
s =
[ ei(-2), ei(-1/2), ei(1), ei(2^(1/2))]

Use vpa to approximate this result with 10-digit accuracy.

vpa(s, 10)
ans =
[ -0.04890051071, -0.5597735948, 1.895117816, 3.048462479]

Branch Cut at Negative Real Axis

The negative real axis is a branch cut. The exponential integral has a jump of height 2 π i when crossing this cut. Compute the exponential integrals at -1, above -1, and below -1 to demonstrate this.

[ei(-1), ei(-1 + 10^(-10)*i), ei(-1 - 10^(-10)*i)]
ans =
  -0.2194 + 0.0000i  -0.2194 + 3.1416i  -0.2194 - 3.1416i

Derivatives of Exponential Integral

Compute the first, second, and third derivatives of a one-argument exponential integral.

syms x
diff(ei(x), x)
diff(ei(x), x, 2)
diff(ei(x), x, 3)
ans =
ans =
exp(x)/x - exp(x)/x^2
ans =
exp(x)/x - (2*exp(x))/x^2 + (2*exp(x))/x^3

Limits of Exponential Integral

Compute the limits of a one-argument exponential integral.

syms x
limit(ei(2*x^2/(1+x)), x, -Inf)
limit(ei(2*x^2/(1+x)), x, 0)
limit(ei(2*x^2/(1+x)), x, Inf)
ans =
ans =
ans =

Input Arguments

collapse all

Input specified as a floating-point number or symbolic number, variable, expression, function, vector, or matrix.


  • The one-argument exponential integral is singular at x = 0. The toolbox uses this special value: ei(0) = -Inf.


The relation between ei and expint is

ei(x) = -expint(1,-x) + (ln(x)-ln(1/x))/2 - ln(-x)

Both functions ei(x) and expint(1,x) have a logarithmic singularity at the origin and a branch cut along the negative real axis. The ei function is not continuous when approached from above or below this branch cut.


[1] Gautschi, W., and W. F. Gahill “Exponential Integral and Related Functions.” Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. (M. Abramowitz and I. A. Stegun, eds.). New York: Dover, 1972.

Version History

Introduced in R2013a

See Also

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