BUILT-IN MATHEMATICA SYMBOL

ExpIntegralEi

ExpIntegralEi[z]
gives the exponential integral function

.

MORE INFORMATION

Mathematical function, suitable for both symbolic and numerical manipulation.

, where the principal value of the integral is taken.

ExpIntegralEi[z] has a branch cut discontinuity in the complex z plane running from to .

For certain special arguments, ExpIntegralEi automatically evaluates to exact values.

ExpIntegralEi can be evaluated to arbitrary numerical precision.

ExpIntegralEi automatically threads over lists.

EXAMPLES

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Basic Examples (3)

Evaluate numerically:

Series expansion around the branch point at the origin:

Evaluate numerically:

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Out[1]=

In[1]:=

Out[1]=

Series expansion around the branch point at the origin:

In[1]:=

Out[1]=

In[2]:=

Out[2]=

Scope (5)

Evaluate to high precision:

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

ExpIntegralEi threads element-wise over lists:

ExpIntegralEi can take complex number inputs:

Simple exact values are generated automatically:

TraditionalForm formatting:

Generalizations & Extensions (2)

ExpIntegralEi can be applied to power series:

Find series expansions at infinity:

Give the result for an arbitrary symbolic direction:

Applications (2)

Compute a classical asymptotic series with k! coefficients:

Plot the imaginary part in the complex plane:

Properties & Relations (8)

Use FullSimplify to simplify expressions containing exponential integrals:

Find the numerical root:

Obtain ExpIntegralEi from integrals and sums:

Calculate limits:

Obtain ExpIntegralEi from a differential equation:

Calculate Wronskian:

Integrals:

Integral transforms:

Possible Issues (3)

ExpIntegralEi can take large values for moderate-size arguments:

ExpIntegralEi has a special value on the negative real axis, not obtained as a limit from either side:

A larger setting for $MaxExtraPrecision can be needed:

Neat Examples (1)

Nested integrals: