# 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 *n*th 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:

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`