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.

The expint 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™.

The expint function fully supports GPU arrays. To run the function on a GPU, specify the input data as a gpuArray (Parallel Computing Toolbox). 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).

Version History

Introduced before R2006a