Evaluate numerically to high precision:

The precision of the output tracks the precision of the input:

ExpIntegralEi can take complex number inputs:

Evaluate ExpIntegralEi efficiently at high precision:

Compute worst-case guaranteed intervals using Interval and CenteredInterval objects:

Or compute average-case statistical intervals using Around:

Compute the elementwise values of an array:

Or compute the matrix ExpIntegralEi function using MatrixFunction:

Value at a fixed point:

Values at infinity:

Find the zero of the ExpIntegralEi:

Plot the ExpIntegralEi function:

Plot the real part of :

Plot the imaginary part of :

ExpIntegralEi is defined for all real values except 0:

Complex domain:

ExpIntegralEi takes all real values:

ExpIntegralEi has the mirror property :

ExpIntegralEi is not an analytic function:

Nor is it meromorphic:

ExpIntegralEi is not monotonic over the reals:

However, it is monotonic over each half-line:

ExpIntegralEi is not injective:

ExpIntegralEi is surjective:

ExpIntegralEi is neither non-negative nor non-positive:

ExpIntegralEi has both singularity and discontinuity at zero:

ExpIntegralEi is neither convex nor concave:

But it is concave over the negative reals:

First derivative:

Higher derivatives:

Formula for the derivative:

Indefinite integral of ExpIntegralEi:

Definite integral of a function involving ExpIntegralEi:

More integrals:

Taylor expansion for ExpIntegralEi around :

Plot the first three approximations for ExpIntegralEi around :

Find series expansion at infinity:

Give the result for an arbitrary symbolic direction:

ExpIntegralEi can be applied to power series:

Use FullSimplify to simplify expressions containing exponential integrals:

Argument simplifications:

For , :

Integral representation:

ExpIntegralEi can be represented as a DifferentialRoot:

ExpIntegralEi can be represented in terms of MeijerG:

TraditionalForm formatting: