# euler

Euler numbers and polynomials

## Description

example

euler(n) returns the nth Euler number.

example

euler(n,x) returns the nth Euler polynomial.

## Examples

### Euler Numbers with Odd and Even Indices

The Euler numbers with even indices alternate the signs. Any Euler number with an odd index is 0.

Compute the even-indexed Euler numbers with the indices from 0 to 10:

euler(0:2:10)
ans =
1          -1           5         -61...
1385      -50521

Compute the odd-indexed Euler numbers with the indices from 1 to 11:

euler(1:2:11)
ans =
0     0     0     0     0     0

### Euler Polynomials

For the Euler polynomials, use euler with two input arguments.

Compute the first, second, and third Euler polynomials in variables x, y, and z, respectively:

syms x y z
euler(1, x)
euler(2, y)
euler(3, z)
ans =
x - 1/2

ans =
y^2 - y

ans =
z^3 - (3*z^2)/2 + 1/4

If the second argument is a number, euler evaluates the polynomial at that number. Here, the result is a floating-point number because the input arguments are not symbolic numbers:

euler(2, 1/3)
ans =
-0.2222

To get the exact symbolic result, convert at least one number to a symbolic object:

euler(2, sym(1/3))
ans =
-2/9

### Plot Euler Polynomials

Plot the first six Euler polynomials.

syms x
fplot(euler(0:5, x), [-1 2])
title('Euler Polynomials')
grid on

### Handle Expressions Containing Euler Polynomials

Many functions, such as diff and expand, can handle expressions containing euler.

Find the first and second derivatives of the Euler polynomial:

syms n x
diff(euler(n,x^2), x)
ans =
2*n*x*euler(n - 1, x^2)
diff(euler(n,x^2), x, x)
ans =
2*n*euler(n - 1, x^2) + 4*n*x^2*euler(n - 2, x^2)*(n - 1)

Expand these expressions containing the Euler polynomials:

expand(euler(n, 2 - x))
ans =
2*(1 - x)^n - (-1)^n*euler(n, x)
expand(euler(n, 2*x))
ans =
(2*2^n*bernoulli(n + 1, x + 1/2))/(n + 1) -...
(2*2^n*bernoulli(n + 1, x))/(n + 1)

## Input Arguments

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Index of the Euler number or polynomial, specified as a nonnegative integer, symbolic nonnegative integer, variable, expression, function, vector, or matrix. If n is a vector or matrix, euler returns Euler numbers or polynomials for each element of n. If one input argument is a scalar and the other one is a vector or a matrix, euler(n,x) expands the scalar into a vector or matrix of the same size as the other argument with all elements equal to that scalar.

Polynomial variable, specified as a symbolic variable, expression, function, vector, or matrix. If x is a vector or matrix, euler returns Euler numbers or polynomials for each element of x. When you use the euler function to find Euler polynomials, at least one argument must be a scalar or both arguments must be vectors or matrices of the same size. If one input argument is a scalar and the other one is a vector or a matrix, euler(n,x) expands the scalar into a vector or matrix of the same size as the other argument with all elements equal to that scalar.

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### Euler Polynomials

The Euler polynomials are defined as follows:

$\frac{2{e}^{xt}}{{e}^{t}+1}=\sum _{n=0}^{\infty }\mathrm{euler}\left(n,x\right)\frac{{t}^{n}}{n!}\text{\hspace{0.17em}}$

### Euler Numbers

The Euler numbers are defined in terms of Euler polynomials as follows:

$\mathrm{euler}\left(n\right)={2}^{n}\mathrm{euler}\left(n,\frac{1}{2}\right)$

## Tips

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

• For the Euler-Mascheroni constant, see eulergamma.