Daily Archives: December 20, 2023

Taylor expansion of f around x

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Series can also be formed by processes such as exponentiation of an
operator or a square matrix. For example, if f is any function of one
argument, and if x and dx are numerical expressions, then this
expression denotes the Taylor expansion of f around x.

(((exp (* dx D)) f) x)
= (+ (f x) (* dx ((D f) x)) (* 1/2 (expt dx 2) (((expt D 2) f) x)) ...)

— refman

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(((exp (* 'dx D)) (literal-function 'f)) 'x)
#| (*series* (0 . 0) (*number* (expression (f x)) (literal-function #[apply-hook 40]) (type-expression Real)) . #[promise 41 (unevaluated)]) |#

(show-expression
 (series:print (((exp (* 'dx D))
                 (literal-function 'f)) 'x)))

(f x)
(* dx ((D f) x))
(* 1/2 (expt dx 2) (((expt D 2) f) x))
(* 1/6 (expt dx 3) (((expt D 3) f) x))
(* 1/24 (expt dx 4) (((expt D 4) f) x))
(* 1/120 (expt dx 5) (((expt D 5) f) x))
(* 1/720 (expt dx 6) (((expt D 6) f) x))
(* 1/5040 (expt dx 7) (((expt D 7) f) x))
(* 1/40320 (expt dx 8) (((expt D 8) f) x))
(* 1/362880 (expt dx 9) (((expt D 9) f) x))
(* 1/3628800 (expt dx 10) (((expt D 10) f) x))
(* 1/39916800 (expt dx 11) (((expt D 11) f) x))
(* 1/479001600 (expt dx 12) (((expt D 12) f) x))
(* 1/6227020800 (expt dx 13) (((expt D 13) f) x))
(* 1/87178291200 (expt dx 14) (((expt D 14) f) x))
;Aborting!: maximum recursion depth exceeded

— Me@2023-12-20 11:30:50 AM

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