Euler numbers and polynomials
euler(
returns the n
)n
th Euler number.
euler(
returns
the n
,x
)n
th Euler polynomial.
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
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 the first six Euler polynomials.
syms x fplot(euler(0:5, x), [-1 2]) title('Euler Polynomials') grid on
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)
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
.