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u t: y Weighted Taylor 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), multiplied by (!y).

(u t: y) differs from (u t. y) by being weighted by the factorial term(s) (!y)

   i=: i.8
   u=: ^         NB. sample u: the exponential function

   u t: i
1 1 1 1 1 1 1 1

Common Uses

Use t: in place of t. to simplify the appearance of the resulting series of coefficients and reveal a hidden pattern. This works because many common Taylor expansions have coefficients with factorial terms in the denominator.

   u t: i
1 1 1 1 1 1 1 1
   u t. i        NB. compare t. with t:
1 1 0.5 0.166667 0.0416667 0.00833333 0.00138889 0.000198413

See Taylor Coefficient (t.) where this same example is discussed in greater depth.

Related Primitives

Taylor Coefficient (t.), Taylor Approximation (T.)