Vocabulary/tdot

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m t. Assign Taylor Coefficient Adverb

No rank -- the result is a verb


u`v t. (i.e.  m t. where m is the gerund u`v) creates a new verb that behaves like u except that its nth Taylor coefficient (as calculated by t.) will be v n.


u t. y Taylor Coefficient Adverb

Rank 0 -- operates on individual atoms of y, producing a result of the same shape -- WHY IS THIS IMPORTANT?



The y-th coefficient(s) in the Taylor series that approximates the function u

Actually, the Maclaurin series. It's always expanded about 0.

The Taylor series for the exponential function is:

Exp.jpg

The coefficients of the first 8 terms of the above series can be written in J like this:

   i=: i.8
   %!i
1 1 0.5 0.166667 0.0416667 0.00833333 0.00138889 0.000198413
   u t. i   NB. for comparison
1 1 0.5 0.166667 0.0416667 0.00833333 0.00138889 0.000198413

Related Primitives

Weighted Taylor (t:), Taylor Approximation (T.)


More Information

1. Only the monadic form of u is considered.

2. u must have an assigned Taylor series using  m t. , or be one of the verbs, or combinations of verbs, for which J knows the Taylor series. These are:

Allowable forms in u t.
Type Allowed Values
constants _9: through 9: _: m"0
monads <: >: +: *: - -. -: ^ [ ] j. o.
bonded dyads m&+ m&* m&- m&^ m&! m&p. +&n *&n -&n %&n ^&n j.&n
circle functions 0&o. (-.&.*:), 1&o. (sin), 2&o. (cos), 5&o. (sinh), 6&o. (cosh), _1&o. (sin^-1^), _3&o. (tan^-1^), _5&o. (sinh^-1^), _7&o. (tanh^-1^)
inverses of the above for all monads except *: ^ ; for bonded dyads except m&^ m&! m&p. ^&n
other inverses %:^:_1 m&%:^:_1 r.&n^:_1
compounds where u and v are allowed u@v u@:v u&v u&:v (u + v) (u * v) (u - v) (u % v)
hooks (+ v) (- v) (* v) (% v)


x u t. y Taylor Term Adverb

Rank 0 0 -- operates on individual atoms of x and y, producing a result of the same shape -- WHY IS THIS IMPORTANT?



The coefficients: u t. y (see above) multiplied by (x^y).

This lets you sum the Taylor series for any given value: x