What is the MacLaurin formula of higher orders for multivariable functions? Example: f(x,y)=cosx cosy

cubanwongux 2022-09-13 Answered
What is the MacLaurin formula of higher orders for multivariable functions? Example: f ( x , y ) = cos x cos y
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Answers (2)

Duncan Kaufman
Answered 2022-09-14 Author has 17 answers
You can expand separately
- cos x = 1 x 2 2 + x 4 4 ! + o ( x 4 )
- cos y = 1 y 2 2 + y 4 4 ! + o ( y 4 )
and then multilply taking the terms to the desidered order.
That is for order IV
f ( x , y ) = cos x cos y = ( 1 x 2 2 + x 4 4 ! + o ( x 4 ) ) ( 1 y 2 2 + y 4 4 ! + o ( y 4 ) ) = = 1 x 2 2 y 2 2 + x 4 4 ! + y 4 4 ! x 2 y 2 4 + o ( | ( x , y ) | 4 )
Note that
- for order II: f ( x , y ) = cos x cos y = 1 x 2 2 y 2 2 + o ( | ( x , y ) | 2 )
- for order III: f ( x , y ) = cos x cos y = 1 x 2 2 y 2 2 + o ( | ( x , y ) | 3 )
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Gauge Odom
Answered 2022-09-15 Author has 4 answers
Your example is special in so far as the function is "separated" into two functions of one variable.
g ( t ) := f ( t x , t y )
of one variable t, and compute its value at t = 1, using the Taylor expansion of a function of one variable:
f ( x , y ) = g ( 1 ) = p = 0 n 1 p ! g ( p ) ( 0 ) + R n   ,
and compute the higher derivatives g ( p ) ( 0 ) using repeatedly the chain rule. Collecting equal terms you obtain
g ( p ) ( 0 ) = k = 0 p ( p k ) f x p k y k ( 0 , 0 ) x p k y k   .
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