expint

Exponential integral function

Syntax

Y = expint(X)

Description

Y = expint(X) evaluates the exponential integral for each element of X.

Examples

Compute Exponential Integral

Find the exponential integral for X = 1+2i.

Y = expint(1+2i)

Y = -0.1268 - 0.0351i

Plot Exponential Integral

Plot the exponential integral for X in the interval of [0,10].

X = 0:0.01:10;
Y = expint(X);
plot(X,Y)
axis([-1 10 -0.5 4])
xlabel('$x$','interpreter','latex')
ylabel('$E_1(x)$','interpreter','latex')
title('Exponential Integral','interpreter','latex')

Input Arguments

X — Input array

scalar | vector | matrix | multidimensional array

Input array, specified as a scalar, vector, matrix, or multidimensional array.

Data Types: single | double
Complex Number Support: Yes

More About

Exponential Integral

The exponential integral of x is defined as

$$E_1(x)={\displaystyle \underset{x}{\overset{\infty }{\int }}{e}^{-t}/tdt}.$$

By analytic continuation, expint is a scalar-valued function in the complex plane cut along the negative real axis.

There is a different function that is sometimes called the exponential integral: the Cauchy principal value integral

$$\text{Ei}(x)={\displaystyle \underset{-\infty }{\overset{x}{\int }}{e}^t/tdt},$$

which, for positive real x, is related to expint as

$$\underset{\delta \to 0+}{\lim }{E}_1(-x+i0)=-\text{Ei}(x)-i\pi .$$

References

[1] Abramowitz, M. and I. A. Stegun. Handbook of Mathematical Functions. Chapter 5, New York: Dover Publications, 1965.

Extended Capabilities

Tall Arrays

Calculate with arrays that have more rows than fit in memory.

This function fully supports tall arrays. For more information, see Tall Arrays.

C/C++ Code Generation
Generate C and C++ code using MATLAB® Coder™.

Thread-Based Environment

Run code in the background using MATLAB® backgroundPool or accelerate code with Parallel Computing Toolbox™ ThreadPool.

This function fully supports thread-based environments. For more information, see Run MATLAB Functions in Thread-Based Environment.

GPU Arrays

Accelerate code by running on a graphics processing unit (GPU) using Parallel Computing Toolbox™.

This function fully supports GPU arrays. For more information, see Run MATLAB Functions on a GPU (Parallel Computing Toolbox).

Distributed Arrays

Partition large arrays across the combined memory of your cluster using Parallel Computing Toolbox™.

This function fully supports distributed arrays. For more information, see Run MATLAB Functions with Distributed Arrays (Parallel Computing Toolbox).