coshint

Hyperbolic cosine integral function

Syntax

coshint(X)

Description

coshint(X) returns the hyperbolic cosine integral function of X.

Examples

Hyperbolic Cosine Integral Function for Numeric and Symbolic Arguments

Depending on its arguments, coshint returns floating-point or exact symbolic results.

Compute the hyperbolic cosine integral function for these numbers. Because these numbers are not symbolic objects, coshint returns floating-point results.

A = coshint([-1, 0, 1/2, 1, pi/2, pi])

A =
   0.8379 + 3.1416i     -Inf + 0.0000i  -0.0528 + 0.0000i   0.8379...
 + 0.0000i   1.7127 + 0.0000i   5.4587 + 0.0000i

Compute the hyperbolic cosine integral function for the numbers converted to symbolic objects. For many symbolic (exact) numbers, coshint returns unresolved symbolic calls.

symA = coshint(sym([-1, 0, 1/2, 1, pi/2, pi]))

symA =
[ coshint(1) + pi*1i, -Inf, coshint(1/2), coshint(1), coshint(pi/2), coshint(pi)]

Use vpa to approximate symbolic results with floating-point numbers:

vpa(symA)

ans =
[ 0.83786694098020824089467857943576...
 + 3.1415926535897932384626433832795i,...
-Inf,...
-0.052776844956493615913136063326141,...
0.83786694098020824089467857943576,...
1.7126607364844281079951569897796,...
5.4587340442160681980014878977798]

Plot Hyperbolic Cosine Integral Function

Plot the hyperbolic cosine integral function on the interval from 0 to 2*pi.

syms x
fplot(coshint(x),[0 2*pi])
grid on

Handle Expressions Containing Hyperbolic Cosine Integral Function

Many functions, such as diff and int, can handle expressions containing coshint.

Find the first and second derivatives of the hyperbolic cosine integral function:

syms x
diff(coshint(x), x)
diff(coshint(x), x, x)

ans =
cosh(x)/x
 
ans =
sinh(x)/x - cosh(x)/x^2

Find the indefinite integral of the hyperbolic cosine integral function:

int(coshint(x), x)

ans =
x*coshint(x) - sinh(x)

Input Arguments

X — Input

symbolic number | symbolic variable | symbolic expression | symbolic function | symbolic vector | symbolic matrix

Input, specified as a symbolic number, variable, expression, or function, or as a vector or matrix of symbolic numbers, variables, expressions, or functions.

More About

Hyperbolic Cosine Integral Function

The hyperbolic cosine integral function is defined as follows:

$$\text{Chi}\left(x\right)=\gamma +\log \left(x\right)+{\displaystyle \underset{0}{\overset{x}{\int }}\frac{\cosh \left(t\right)-1}{t}}dt$$

Here, γ is the Euler–Mascheroni constant:

$$\gamma =\underset{n\to \infty }{\lim }\left(\left({\displaystyle \sum _{k=1}^n\frac{1}{k}}\right)-\ln \left(n\right)\right)$$

References

[1] Cautschi, W. and W. F. Cahill. “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 R2014a