Editing Taylor series (section)

==Taylor series in several variables{{anchor|In several variables}}==

The Taylor series may also be generalized to functions of more than one variable with<ref>{{multiref|{{harvnb|Hörmander|2002|loc=See Eqq. 1.1.7 and 1.1.7′}}
 |{{harvnb|Kolk|Duistermaat|2010|p=59–63}}
}}</ref>

<math display="block">\begin{align}
T(x_1,\ldots,x_d) &= \sum_{n_1=0}^\infty \cdots \sum_{n_d = 0}^\infty  \frac{(x_1-a_1)^{n_1}\cdots (x_d-a_d)^{n_d}}{n_1!\cdots n_d!}\,\left(\frac{\partial^{n_1 + \cdots + n_d}f}{\partial x_1^{n_1}\cdots \partial x_d^{n_d}}\right)(a_1,\ldots,a_d) \\
&= f(a_1, \ldots,a_d) + \sum_{j=1}^d \frac{\partial f(a_1, \ldots,a_d)}{\partial x_j} (x_j - a_j) + \frac{1}{2!} \sum_{j=1}^d \sum_{k=1}^d \frac{\partial^2 f(a_1, \ldots,a_d)}{\partial x_j \partial x_k} (x_j - a_j)(x_k - a_k) \\ 
& \qquad \qquad + \frac{1}{3!} \sum_{j=1}^d\sum_{k=1}^d\sum_{l=1}^d \frac{\partial^3 f(a_1, \ldots,a_d)}{\partial x_j \partial x_k \partial x_l} (x_j - a_j)(x_k - a_k)(x_l - a_l) + \cdots
\end{align}</math>

For example, for a function <math>f(x,y)</math> that depends on two variables, {{mvar|x}} and {{mvar|y}}, the Taylor series to second order about the point {{math|(''a'', ''b'')}} is

<math display="block">f(a,b) +(x-a) f_x(a,b) +(y-b) f_y(a,b) + \frac{1}{2!}\Big( (x-a)^2 f_{xx}(a,b) + 2(x-a)(y-b) f_{xy}(a,b) +(y-b)^2 f_{yy}(a,b) \Big)</math>

where the subscripts denote the respective [[partial derivative]]s.

===Second-order Taylor series in several variables===
{{see also|Linearization#Multivariable functions}}

A second-order Taylor series expansion of a scalar-valued function of more than one variable can be written compactly as

<math display="block">T(\mathbf{x}) = f(\mathbf{a}) + (\mathbf{x} - \mathbf{a})^\mathsf{T} D f(\mathbf{a}) + \frac{1}{2!} (\mathbf{x} - \mathbf{a})^\mathsf{T} \left \{D^2 f(\mathbf{a}) \right \} (\mathbf{x} - \mathbf{a}) + \cdots,</math>

where {{math|''D'' ''f''&thinsp;('''a''')}} is the [[gradient]] of {{mvar|f}} evaluated at {{math|'''x''' {{=}} '''a'''}} and {{math|''D''<sup>2</sup> ''f''&thinsp;('''a''')}} is the [[Hessian matrix]]. Applying the [[multi-index notation]] the Taylor series for several variables becomes

<math display="block">T(\mathbf{x}) = \sum_{|\alpha| \geq 0}\frac{(\mathbf{x}-\mathbf{a})^\alpha}{\alpha !} \left({\mathrm{\partial}^{\alpha}}f\right)(\mathbf{a}),</math>

which is to be understood as a still more abbreviated [[multi-index]] version of the first equation of this paragraph, with a full analogy to the single variable case.

=== Example ===
[[Image:Second Order Taylor.svg|200px|thumb|right|Second-order Taylor series approximation (in orange) of a function {{math|''f''&thinsp;(''x'',''y'') {{=}} ''e<sup>x</sup>'' ln(1 + ''y'')}} around the origin.]]
In order to compute a second-order Taylor series expansion around point {{math|(''a'', ''b'') {{=}} (0, 0)}} of the function
<math display="block">f(x,y)=e^x\ln(1+y),</math>

one first computes all the necessary partial derivatives:

<math display="block">\begin{align}
f_x &= e^x\ln(1+y) \\[6pt]
f_y &= \frac{e^x}{1+y} \\[6pt]
f_{xx} &= e^x\ln(1+y) \\[6pt]
f_{yy}  &= - \frac{e^x}{(1+y)^2}  \\[6pt]
f_{xy} &=f_{yx} =  \frac{e^x}{1+y} .
\end{align}</math>

Evaluating these derivatives at the origin gives the Taylor coefficients

<math display="block">\begin{align}
f_x(0,0) &= 0 \\
f_y(0,0) &=1 \\
f_{xx}(0,0) &=0 \\
f_{yy}(0,0) &=-1 \\
f_{xy}(0,0) &=f_{yx}(0,0)=1.
\end{align}</math>

Substituting these values in to the general formula

<math display="block">\begin{align}
T(x,y) = &f(a,b) +(x-a) f_x(a,b) +(y-b) f_y(a,b) 
\\ &{}+\frac{1}{2!}\left( (x-a)^2f_{xx}(a,b) 
+ 2(x-a)(y-b)f_{xy}(a,b) +(y-b)^2 f_{yy}(a,b) \right)+ \cdots
\end{align}</math>

produces

<math display="block">\begin{align}
T(x,y) &= 0 + 0(x-0) + 1(y-0) + \frac{1}{2}\big( 0(x-0)^2 + 2(x-0)(y-0) + (-1)(y-0)^2 \big) + \cdots \\
&= y + xy - \tfrac12 y^2 + \cdots 
\end{align}</math>

Since {{math|ln(1 + ''y'')}} is analytic in {{math|{{abs|''y''}} < 1}}, we have

<math display="block">e^x\ln(1+y)= y + xy - \tfrac12 y^2 + \cdots, \qquad |y| < 1.</math>