multivariate Taylor series expansion - Maple Help

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mtaylor - multivariate Taylor series expansion

Calling Sequence

mtaylor(f, v)

mtaylor(f, v, n)

mtaylor(f, v, n, w)

Parameters

f

-

algebraic expression

v

-

list or set of names or equations

n

-

(optional) non-negative integer

w

-

(optional) list of positive integers

Description

• 

The mtaylor function computes a truncated multivariate Taylor series expansion of the input expression f, with respect to the variables v, to order n, using the variable weights w.

• 

The variables v can be a list or set of names or equations. If vi is an equation, then the left-hand side of vi is the variable, and the right-hand side is the point of expansion.  If vi is a name, then vi=0 is assumed as the point of expansion.

• 

If the third argument n is present then it specifies the ``truncation order'' of the series. The concept of ``truncation order'' used is ``total degree'' in the variables. If n is not present, the truncation order used is the value of the global variable Order, which is 6 by default.

• 

If the fourth argument w is present it specifies the variable weights to be used (by default all 1).  A weight of 2 will halve the order in the corresponding variable to which the series is computed.

• 

Note:  mtaylor restricts its domain to ``pure'' Taylor series, those series with non-negative powers in the variables.

Examples

mtaylorⅇx2+y2,x,y,8

1+x2+y2+12x4+y2x2+12y4+16x6+12y2x4+12y4x2+16y6

(1)

mtaylor1+x2+y2,x,y,8

1+12x2+12y218x414y2x218y4+116x6+316y2x4+316y4x2+116y6

(2)

mtaylorsinx2+y2,x,y

x2+y2

(3)

mtaylorsinx2+y2,x,y,8

x2+y216x612y2x412y4x216y6

(4)

mtaylorsinx2+y2,x,y,8,2,1

y2+x216y6

(5)

mtaylorsinx2+y2,x=1,y,3

sin1+2cos1x1+2sin1+cos1x12+cos1y2

(6)

mtaylorcosx2+y2,x=1,y=2,3

cos52sin5x14sin5y2+2cos5sin5x128cos5y2x1+8cos5sin5y22

(7)

evalf

9.305580560+1.917848549x+3.835697099y+0.3915999037x1.22.269297484y2.x1.1.310373209y2.2

(8)

mtaylorfx,y,x,y,3

f0,0+D1f0,0x+D2f0,0y+12D1,1f0,0x2+D1,2f0,0xy+12D2,2f0,0y2

(9)

See Also

coeftayl, poisson, series, taylor


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