eulergamma

Euler-Mascheroni constant

Syntax

eulergamma

Description

eulergamma represents the Euler-Mascheroni constant. To get a floating-point approximation with the current precision set by digits, use vpa(eulergamma).

Examples

Represent and Numerically Approximate the Euler-Mascheroni Constant

Represent the Euler-Mascheroni constant using eulergamma, which returns the symbolic form eulergamma.

eulergamma

ans =
eulergamma

Use eulergamma in symbolic calculations. Numerically approximate your result with vpa.

a = eulergamma;
g = a^2 + log(a)
gVpa = vpa(g)

g =
log(eulergamma) + eulergamma^2
gVpa =
-0.21636138917392614801928563244766

Find the double-precision approximation of the Euler-Mascheroni constant using double.

double(eulergamma)

ans =
    0.5772

Show Relation of Euler-Mascheroni Constant to Gamma Functions

Show the relations between the Euler-Mascheroni constant γ, digamma function Ψ, and gamma function Γ.

Show that $$\gamma =-\Psi \left(1\right)$$.

-psi(sym(1))

ans =
eulergamma

Show that $$\gamma =-\Gamma \text{'}\left(x\right)|{}_{x=1}$$.

syms x
-subs(diff(gamma(x)),x,1)

ans =
eulergamma

More About

Euler-Mascheroni Constant

The Euler-Mascheroni constant is defined as follows:

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

Tips

• For the value e = 2.71828…, called Euler’s number, use exp(1) to return the double-precision representation. For the exact representation of Euler’s number e, call exp(sym(1)).

• For the other meaning of Euler’s numbers and for Euler’s polynomials, see euler.