Taylor Series

series(cs, x, [x0=0,], ns=0:length(cs)-1)

Return the power series in x around x0 to the powers ns with coefficients cs.

series(y, x, [x0=0,] ns)

Return the power series in x around x0 to the powers ns with coefficients automatically created from the variable y.

Examples

julia> @variables x y[0:3] z
3-element Vector{Any}:
 x
  y[0:3]
 z

julia> series(y, x, 2)
y[0] + (-2 + x)*y[1] + ((-2 + x)^2)*y[2] + ((-2 + x)^3)*y[3]

julia> series(z, x, 2, 0:3)
z[0] + (-2 + x)*z[1] + ((-2 + x)^2)*z[2] + ((-2 + x)^3)*z[3]

taylor(f, x, [x0=0,] n; rationalize=true, kwargs...)

Calculate the n-th order term(s) in the Taylor series of f around x = x0. If rationalize, float coefficients are approximated as rational numbers (this can produce unexpected results for irrational numbers, for example). Keyword arguments kwargs... are forwarded to internal substitute() calls.

Examples

julia> @variables x
1-element Vector{Num}:
 x

julia> taylor(exp(x), x, 0:3)
1 + x + (1//2)*(x^2) + (1//6)*(x^3)

julia> taylor(exp(x), x, 0:3; rationalize=false)
1.0 + x + 0.5(x^2) + 0.16666666666666666(x^3)

julia> taylor(√(x), x, 1, 0:3)
1 + (1//2)*(-1 + x) - (1//8)*((-1 + x)^2) + (1//16)*((-1 + x)^3)

julia> isequal(taylor(exp(im*x), x, 0:5), taylor(exp(im*x), x, 0:5))
true

taylor_coeff(f, x[, n]; rationalize=true, kwargs...)

Calculate the n-th order coefficient(s) in the Taylor series of f around x = 0. If rationalize, float coefficients are approximated as rational numbers (this can produce unexpected results for irrational numbers, for example). Keyword arguments kwargs... are forwarded to internal substitute() calls.

Examples

julia> @variables x y
2-element Vector{Num}:
 x
 y

julia> taylor_coeff(series(y, x, 0:5), x, 0:2:4)
3-element Vector{Num}:
 y[0]
 y[2]
 y[4]