§3.4 Differentiation

Referenced by:: §3.3(v)
Permalink:: http://dlmf.nist.gov/3.4

§3.4(i) Equally-Spaced Nodes
§3.4(ii) Analytic Functions
§3.4(iii) Partial Derivatives

§3.4(i) Equally-Spaced Nodes

Notes:: See Hildebrand (1974, pp. 85–89). The coefficients $B_{k}^{n}$ are obtained by differentiation of the $A_{k}^{n}$ ; compare (3.4.2).
Keywords:: differentiation
Permalink:: http://dlmf.nist.gov/3.4.i

The Lagrange $(n+1)$ -point formula is

3.4.1 $hf^{{\prime}}_{t}=hf^{{\prime}}(x_{0}+th)=\sum _{{k=n_{0}}}^{{n_{1}}}B_{k}^{n}f_{k}+hR^{{\prime}}_{{n,t}},$ $n_{0}<t<n_{1}$ ,

Symbols:: $B_{k}^{n}$ : differentiated Lagrangian interpolation coefficients and $R^{{\prime}}_{{n,t}}(x)$ : remainder
Referenced by:: §3.4(i)
Permalink:: http://dlmf.nist.gov/3.4.E1
Encodings:: TeX, pMML, png

and follows from the differentiated form of (3.3.4). The $B_{k}^{n}$ are the differentiated Lagrangian interpolation coefficients:

3.4.2 $B_{k}^{n}=\ifrac{dA_{k}^{n}}{dt},$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $A_{k}^{n}$ : Lagrangian interpolation coefficients and $B_{k}^{n}$ : differentiated Lagrangian interpolation coefficients
Referenced by:: §3.4(i)
Permalink:: http://dlmf.nist.gov/3.4.E2
Encodings:: TeX, pMML, png

where $A_{k}^{n}$ is as in (3.3.10).

If $f^{{(n+2)}}(x)$ is continuous on the interval $I$ defined in §3.3(i), then the remainder in (3.4.1) is given by

3.4.3 $hR^{{\prime}}_{{n,t}}=\frac{h^{{n+1}}}{(n+1)!}\left(f^{{(n+1)}}(\xi _{0})\frac{d}{dt}\prod _{{k=n_{0}}}^{{n_{1}}}(t-k)+f^{{(n+2)}}(\xi _{1})\prod _{{k=n_{0}}}^{{n_{1}}}(t-k)\right),$

Symbols:: $\frac{df}{dx}$ : derivative of $f$ with respect to $x$ , $!$ : $n!$ : factorial and $R^{{\prime}}_{{n,t}}(x)$ : remainder
Permalink:: http://dlmf.nist.gov/3.4.E3
Encodings:: TeX, pMML, png

where $\xi _{0}$ and $\xi _{1}\in I$ .

For the values of $n_{0}$ and $n_{1}$ used in the formulas below

3.4.4 $h\left|R^{{\prime}}_{{n,t}}\right|\leq h^{{n+1}}\left(c_{n}\left|f^{{(n+2)}}(\xi _{1})\right|+\frac{1}{n+1}\left|f^{{(n+1)}}(\xi _{0})\right|\right),$ $n_{0}<t<n_{1}$ ,

Symbols:: $c_{n}$ : product and $R^{{\prime}}_{{n,t}}(x)$ : remainder
Permalink:: http://dlmf.nist.gov/3.4.E4
Encodings:: TeX, pMML, png

where $c_{n}$ is defined by (3.3.12), with numerical values as in §3.3(ii).

¶ Two-Point Formula

3.4.5 $hf^{{\prime}}_{t}=-f_{0}+f_{1}+hR^{{\prime}}_{{1,t}},$ $0<t<1$ .

Symbols:: $R^{{\prime}}_{{n,t}}(x)$ : remainder
Permalink:: http://dlmf.nist.gov/3.4.E5
Encodings:: TeX, pMML, png

¶ Three-Point Formula

3.4.6 $hf^{{\prime}}_{t}=-\tfrac{1}{2}(1-2t)f_{{-1}}-2tf_{0}+\tfrac{1}{2}(1+2t)f_{1}+hR^{{\prime}}_{{2,t}},$ $\left|t\right|<1$ .

Symbols:: $R^{{\prime}}_{{n,t}}(x)$ : remainder
A&S Ref:: 25.3.4
Permalink:: http://dlmf.nist.gov/3.4.E6
Encodings:: TeX, pMML, png

¶ Four-Point Formula

3.4.7 $hf^{{\prime}}_{t}=\sum _{{k=-1}}^{2}B_{k}^{3}f_{k}+hR^{{\prime}}_{{3,t}},$ $-1<t<2$ ,

Symbols:: $B_{k}^{n}$ : differentiated Lagrangian interpolation coefficients and $R^{{\prime}}_{{n,t}}(x)$ : remainder
A&S Ref:: 25.3.5 (modification of)
Permalink:: http://dlmf.nist.gov/3.4.E7
Encodings:: TeX, pMML, png

3.4.8

$B_{{-1}}^{3}=-\tfrac{1}{6}(2-6t+3t^{2}),$

$B_{0}^{3}=-\tfrac{1}{2}(1+4t-3t^{2}),$

$B_{1}^{3}=\tfrac{1}{2}(2+2t-3t^{2}),$

$B_{2}^{3}=-\tfrac{1}{6}(1-3t^{2}).$

Symbols:: $B_{k}^{n}$ : differentiated Lagrangian interpolation coefficients
Permalink:: http://dlmf.nist.gov/3.4.E8
Encodings:: TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, png, png, png, png

¶ Five-Point Formula

3.4.9 $hf^{{\prime}}_{t}=\sum _{{k=-2}}^{2}B_{k}^{4}f_{k}+hR^{{\prime}}_{{4,t}},$ $\left|t\right|<2$ ,

Symbols:: $B_{k}^{n}$ : differentiated Lagrangian interpolation coefficients and $R^{{\prime}}_{{n,t}}(x)$ : remainder
A&S Ref:: 25.3.6 (modification of)
Permalink:: http://dlmf.nist.gov/3.4.E9
Encodings:: TeX, pMML, png

3.4.10

$B_{{-2}}^{4}=\tfrac{1}{12}(1-t-3t^{2}+2t^{3}),$

$B_{{-1}}^{4}=-\tfrac{1}{6}(4-8t-3t^{2}+4t^{3}),$

$B_{0}^{4}=-\tfrac{1}{2}t(5-2t^{2}),$

$B_{1}^{4}=\tfrac{1}{6}(4+8t-3t^{2}-4t^{3}),$

$B_{2}^{4}=-\tfrac{1}{12}(1+t-3t^{2}-2t^{3}).$

Symbols:: $B_{k}^{n}$ : differentiated Lagrangian interpolation coefficients
Permalink:: http://dlmf.nist.gov/3.4.E10
Encodings:: TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png

¶ Six-Point Formula

3.4.11 $hf^{{\prime}}_{t}=\sum _{{k=-2}}^{3}B_{k}^{5}f_{k}+hR^{{\prime}}_{{5,t}},$ $-2<t<3$ ,

Symbols:: $B_{k}^{n}$ : differentiated Lagrangian interpolation coefficients and $R^{{\prime}}_{{n,t}}(x)$ : remainder
Permalink:: http://dlmf.nist.gov/3.4.E11
Encodings:: TeX, pMML, png

3.4.12

$B_{{-2}}^{5}=\tfrac{1}{120}(6-10t-15t^{2}+20t^{3}-5t^{4}),$

$B_{{-1}}^{5}=-\tfrac{1}{24}(12-32t+3t^{2}+16t^{3}-5t^{4}),$

$B_{0}^{5}=-\tfrac{1}{12}(4+30t-15t^{2}-12t^{3}+5t^{4}),$

$B_{1}^{5}=\tfrac{1}{12}(12+16t-21t^{2}-8t^{3}+5t^{4}),$

$B_{2}^{5}=-\tfrac{1}{24}(6+2t-21t^{2}-4t^{3}+5t^{4}),$

$B_{3}^{5}=\tfrac{1}{120}(4-15t^{2}+5t^{4}).$

Symbols:: $B_{k}^{n}$ : differentiated Lagrangian interpolation coefficients
Permalink:: http://dlmf.nist.gov/3.4.E12
Encodings:: TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png

¶ Seven-Point Formula

3.4.13 $hf^{{\prime}}_{t}=\sum _{{k=-3}}^{3}B_{k}^{6}f_{k}+hR^{{\prime}}_{{6,t}},$ $\left|t\right|<3$ ,

Symbols:: $B_{k}^{n}$ : differentiated Lagrangian interpolation coefficients and $R^{{\prime}}_{{n,t}}(x)$ : remainder
Permalink:: http://dlmf.nist.gov/3.4.E13
Encodings:: TeX, pMML, png

3.4.14

$B_{{-3}}^{6}=-\tfrac{1}{720}(12-8t-45t^{2}+20t^{3}+15t^{4}-6t^{5}),$

$B_{{-2}}^{6}=\tfrac{1}{60}(9-9t-30t^{2}+20t^{3}+5t^{4}-3t^{5}),$

$B_{{-1}}^{6}=-\tfrac{1}{48}(36-72t-39t^{2}+52t^{3}+5t^{4}-6t^{5}),$

$B_{0}^{6}=-\tfrac{1}{18}t(49-28t^{2}+3t^{4}),$

$B_{1}^{6}=\tfrac{1}{48}(36+72t-39t^{2}-52t^{3}+5t^{4}+6t^{5}),$

$B_{2}^{6}=-\tfrac{1}{60}(9+9t-30t^{2}-20t^{3}+5t^{4}+3t^{5}),$

$B_{3}^{6}=\tfrac{1}{720}(12+8t-45t^{2}-20t^{3}+15t^{4}+6t^{5}).$

Symbols:: $B_{k}^{n}$ : differentiated Lagrangian interpolation coefficients
Permalink:: http://dlmf.nist.gov/3.4.E14
Encodings:: TeX, TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png, png

¶ Eight-Point Formula

Keywords:: differentiation

3.4.15 $hf^{{\prime}}_{t}=\sum _{{k=-3}}^{4}B_{k}^{7}f_{k}+hR^{{\prime}}_{{7,t}},$ $-3<t<4$ ,

Symbols:: $B_{k}^{n}$ : differentiated Lagrangian interpolation coefficients and $R^{{\prime}}_{{n,t}}(x)$ : remainder
Permalink:: http://dlmf.nist.gov/3.4.E15
Encodings:: TeX, pMML, png

3.4.16

$B_{{-3}}^{7}=-\tfrac{1}{5040}(48-56t-168t^{2}+140t^{3}+35t^{4}-42t^{5}+7t^{6}),$

$B_{{-2}}^{7}=\tfrac{1}{720}(72-108t-213t^{2}+240t^{3}-10t^{4}-36t^{5}+7t^{6}),$

$B_{{-1}}^{7}=-\tfrac{1}{240}(144-360t-48t^{2}+260t^{3}-45t^{4}-30t^{5}+7t^{6}),$

$B_{0}^{7}=-\tfrac{1}{144}(36+392t-147t^{2}-224t^{3}+70t^{4}+24t^{5}-7t^{6}),$

$B_{1}^{7}=\tfrac{1}{144}(144+216t-264t^{2}-156t^{3}+85t^{4}+18t^{5}-7t^{6}),$

$B_{2}^{7}=-\tfrac{1}{240}(72+36t-267t^{2}-80t^{3}+90t^{4}+12t^{5}-7t^{6}),$

$B_{3}^{7}=\tfrac{1}{720}(48+8t-192t^{2}-20t^{3}+85t^{4}+6t^{5}-7t^{6}),$

$B_{4}^{7}=-\tfrac{1}{5040}(36-147t^{2}+70t^{4}-7t^{6}).$

Symbols:: $B_{k}^{n}$ : differentiated Lagrangian interpolation coefficients
Permalink:: http://dlmf.nist.gov/3.4.E16
Encodings:: TeX, TeX, TeX, TeX, TeX, TeX, TeX, TeX, pMML, pMML, pMML, pMML, pMML, pMML, pMML, pMML, png, png, png, png, png, png, png, png

For corresponding formulas for second, third, and fourth derivatives, with $t=0$ , see Collatz (1960, Table III, pp. 538–539). For formulas for derivatives with equally-spaced real nodes and based on Sinc approximations (§3.3(vi)), see Stenger (1993, §3.5).

§3.4(ii) Analytic Functions

Keywords:: differentiation
Permalink:: http://dlmf.nist.gov/3.4.ii

If $f$ can be extended analytically into the complex plane, then from Cauchy’s integral formula (§1.9(iii))

3.4.17 $\frac{1}{k!}\, f^{{(k)}}(x_{0})=\frac{1}{2\pi i}\int _{C}\frac{f(\zeta)}{(\zeta-x_{0})^{{k+1}}}\, d\zeta,$

Symbols:: $dx$ : differential of $x$ , $!$ : $n!$ : factorial, $\int$ : integral and $C$ : simple closed contour
Permalink:: http://dlmf.nist.gov/3.4.E17
Encodings:: TeX, pMML, png

where $C$ is a simple closed contour described in the positive rotational sense such that $C$ and its interior lie in the domain of analyticity of $f$ , and $x_{0}$ is interior to $C$ . Taking $C$ to be a circle of radius $r$ centered at $x_{0}$ , we obtain

3.4.18 $\frac{1}{k!}\, f^{{(k)}}(x_{0})=\frac{1}{2\pi r^{k}}\int _{0}^{{2\pi}}f(x_{0}+re^{{i\theta}})e^{{-ik\theta}}d\theta.$

Symbols:: $dx$ : differential of $x$ , $e$ : base of exponential function, $!$ : $n!$ : factorial, $\int$ : integral and $r$ : radius
Referenced by:: ¶ ‣ §3.4(ii)
Permalink:: http://dlmf.nist.gov/3.4.E18
Encodings:: TeX, pMML, png

The integral on the right-hand side can be approximated by the composite trapezoidal rule (3.5.2).

¶ Example

$f(z)=e^{z}$ , $x_{0}=0$ . The integral (3.4.18) becomes

3.4.19 $\frac{1}{k!}=\frac{1}{2\pi r^{k}}\int _{0}^{{2\pi}}e^{{r\mathop{\cos\/}\nolimits\theta}}\mathop{\cos\/}\nolimits\!\left(r\mathop{\sin\/}\nolimits\theta-k\theta\right)d\theta.$

Symbols:: $\mathop{\cos\/}\nolimits z$ : cosine function, $dx$ : differential of $x$ , $e$ : base of exponential function, $!$ : $n!$ : factorial, $\int$ : integral, $\mathop{\sin\/}\nolimits z$ : sine function and $r$ : radius
Permalink:: http://dlmf.nist.gov/3.4.E19
Encodings:: TeX, pMML, png

With the choice $r=k$ (which is crucial when $k$ is large because of numerical cancellation) the integrand equals $e^{k}$ at the dominant points $\theta=0,2\pi$ , and in combination with the factor $k^{{-k}}$ in front of the integral sign this gives a rough approximation to $1/k!$ . The choice $r=k$ is motivated by saddle-point analysis; see §2.4(iv) or examples in §3.5(ix). As explained in §§3.5(i) and 3.5(ix) the composite trapezoidal rule can be very efficient for computing integrals with analytic periodic integrands.

§3.4(iii) Partial Derivatives

Keywords:: differentiation
Permalink:: http://dlmf.nist.gov/3.4.iii

¶ First-Order

For partial derivatives we use the notation $u_{{t,s}}=u(x_{0}+th,y_{0}+sh)$ .

3.4.20 $\frac{\partial u_{{0,0}}}{\partial x}=\frac{1}{2h}\,(u_{{1,0}}-u_{{-1,0}})+\mathop{O\/}\nolimits\!\left(h^{2}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ and $u$ : function
A&S Ref:: 25.3.21
Permalink:: http://dlmf.nist.gov/3.4.E20
Encodings:: TeX, pMML, png

3.4.21 $\frac{\partial u_{{0,0}}}{\partial x}=\frac{1}{4h}\,(u_{{1,1}}-u_{{-1,1}}+u_{{1,-1}}-u_{{-1,-1}})+\mathop{O\/}\nolimits\!\left(h^{2}\right).$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ and $u$ : function
A&S Ref:: 25.3.22
Permalink:: http://dlmf.nist.gov/3.4.E21
Encodings:: TeX, pMML, png

¶ Second-Order

3.4.22 $\frac{{\partial}^{2}u_{{0,0}}}{{\partial x}^{2}}=\frac{1}{h^{2}}\,(u_{{1,0}}-2u_{{0,0}}+u_{{-1,0}})+\mathop{O\/}\nolimits\!\left(h^{2}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ and $u$ : function
A&S Ref:: 25.3.23
Permalink:: http://dlmf.nist.gov/3.4.E22
Encodings:: TeX, pMML, png

3.4.23 $\frac{{\partial}^{2}u_{{0,0}}}{{\partial x}^{2}}=\frac{1}{12h^{2}}\,(-u_{{2,0}}+16u_{{1,0}}-30u_{{0,0}}+16u_{{-1,0}}-u_{{-2,0}})+\mathop{O\/}\nolimits\!\left(h^{4}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ and $u$ : function
A&S Ref:: 25.3.24
Permalink:: http://dlmf.nist.gov/3.4.E23
Encodings:: TeX, pMML, png

3.4.24 $\frac{{\partial}^{2}u_{{0,0}}}{{\partial x}^{2}}=\frac{1}{3h^{2}}\,(u_{{1,1}}-2u_{{0,1}}+u_{{-1,1}}+u_{{1,0}}-2u_{{0,0}}+u_{{-1,0}}+u_{{1,-1}}-2u_{{0,-1}}+u_{{-1,-1}})+\mathop{O\/}\nolimits\!\left(h^{2}\right).$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ and $u$ : function
A&S Ref:: 25.3.25
Permalink:: http://dlmf.nist.gov/3.4.E24
Encodings:: TeX, pMML, png

3.4.25 $\frac{{\partial}^{2}u_{{0,0}}}{\partial x\partial y}=\frac{1}{4h^{2}}\,(u_{{1,1}}-u_{{1,-1}}-u_{{-1,1}}+u_{{-1,-1}})+\mathop{O\/}\nolimits\!\left(h^{2}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\partial x$ : partial differential of $x$ and $u$ : function
A&S Ref:: 25.3.26
Permalink:: http://dlmf.nist.gov/3.4.E25
Encodings:: TeX, pMML, png

3.4.26 $\frac{{\partial}^{2}u_{{0,0}}}{\partial x\partial y}=-\frac{1}{2h^{2}}\,(u_{{1,0}}+u_{{-1,0}}+u_{{0,1}}+u_{{0,-1}}-2u_{{0,0}}-u_{{1,1}}-u_{{-1,-1}})+\mathop{O\/}\nolimits\!\left(h^{2}\right).$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\partial x$ : partial differential of $x$ and $u$ : function
A&S Ref:: 25.3.27
Permalink:: http://dlmf.nist.gov/3.4.E26
Encodings:: TeX, pMML, png

¶ Laplacian

Keywords:: Laplacian

3.4.27 $\nabla^{2}u=\frac{{\partial}^{2}u}{{\partial x}^{2}}+\frac{{\partial}^{2}u}{{\partial y}^{2}}\,.$

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ and $u$ : function
A&S Ref:: 25.3.30
Permalink:: http://dlmf.nist.gov/3.4.E27
Encodings:: TeX, pMML, png

3.4.28 $\nabla^{2}u_{{0,0}}=\frac{1}{h^{2}}\,(u_{{1,0}}+u_{{0,1}}+u_{{-1,0}}+u_{{0,-1}}-4u_{{0,0}})+\mathop{O\/}\nolimits\!\left(h^{2}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding and $u$ : function
A&S Ref:: 25.3.30
Permalink:: http://dlmf.nist.gov/3.4.E28
Encodings:: TeX, pMML, png

3.4.29 $\nabla^{2}u_{{0,0}}=\frac{1}{12h^{2}}\left(-60u_{{0,0}}+16(u_{{1,0}}+u_{{0,1}}+u_{{-1,0}}+u_{{0,-1}})-(u_{{2,0}}+u_{{0,2}}+u_{{-2,0}}+u_{{0,-2}})\right)+\mathop{O\/}\nolimits\!\left(h^{4}\right).$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding and $u$ : function
A&S Ref:: 25.3.31
Permalink:: http://dlmf.nist.gov/3.4.E29
Encodings:: TeX, pMML, png

¶ Fourth-Order

3.4.30 $\frac{{\partial}^{4}u_{{0,0}}}{{\partial x}^{4}}=\frac{1}{h^{4}}\,(u_{{2,0}}-4u_{{1,0}}+6u_{{0,0}}-4u_{{-1,0}}+u_{{-2,0}})+\mathop{O\/}\nolimits\!\left(h^{2}\right).$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ and $u$ : function
A&S Ref:: 25.3.28
Permalink:: http://dlmf.nist.gov/3.4.E30
Encodings:: TeX, pMML, png

3.4.31 $\frac{{\partial}^{4}u_{{0,0}}}{{\partial}^{2}x{\partial}^{2}y}=\frac{1}{h^{4}}\,(u_{{1,1}}+u_{{-1,1}}+u_{{1,-1}}+u_{{-1,-1}}-2u_{{1,0}}-2u_{{-1,0}}-2u_{{0,1}}-2u_{{0,-1}}+4u_{{0,0}})+\mathop{O\/}\nolimits\!\left(h^{2}\right).$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding, $\partial x$ : partial differential of $x$ and $u$ : function
A&S Ref:: 25.3.29
Permalink:: http://dlmf.nist.gov/3.4.E31
Encodings:: TeX, pMML, png

¶ Biharmonic Operator

Keywords:: biharmonic operator, differentiation

3.4.32 $\nabla^{4}u=\frac{{\partial}^{4}u}{{\partial x}^{4}}+2\frac{{\partial}^{4}u}{{\partial}^{2}x{\partial}^{2}y}+\frac{{\partial}^{4}u}{{\partial y}^{4}}\,.$

Symbols:: $\frac{\partial f}{\partial x}$ : partial derivative of $f$ with respect to $x$ , $\partial x$ : partial differential of $x$ and $u$ : function
A&S Ref:: 25.3.32
Permalink:: http://dlmf.nist.gov/3.4.E32
Encodings:: TeX, pMML, png

3.4.33 $\nabla^{4}u_{{0,0}}=\frac{1}{h^{4}}\,(20u_{{0,0}}-8(u_{{1,0}}+u_{{0,1}}+u_{{-1,0}}+u_{{0,-1}})+2(u_{{1,1}}+u_{{1,-1}}+u_{{-1,1}}+u_{{-1,-1}})+(u_{{0,2}}+u_{{2,0}}+u_{{-2,0}}+u_{{0,-2}}))+\mathop{O\/}\nolimits\!\left(h^{2}\right),$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding and $u$ : function
A&S Ref:: 25.3.32
Permalink:: http://dlmf.nist.gov/3.4.E33
Encodings:: TeX, pMML, png

3.4.34 $\nabla^{4}u_{{0,0}}=\frac{1}{6h^{4}}\,(184u_{{0,0}}-(u_{{0,3}}+u_{{0,-3}}+u_{{3,0}}+u_{{-3,0}})+14(u_{{0,2}}+u_{{0,-2}}+u_{{2,0}}+u_{{-2,0}})-77(u_{{0,1}}+u_{{0,-1}}+u_{{1,0}}+u_{{-1,0}})+20(u_{{1,1}}+u_{{1,-1}}+u_{{-1,1}}+u_{{-1,-1}})-(u_{{1,2}}+u_{{2,1}}+u_{{1,-2}}+u_{{2,-1}}+u_{{-1,2}}+u_{{-2,1}}+u_{{-1,-2}}+u_{{-2,-1}}))+\mathop{O\/}\nolimits\!\left(h^{4}\right).$

Symbols:: $\mathop{O\/}\nolimits\!\left(x\right)$ : order not exceeding and $u$ : function
A&S Ref:: 25.3.33
Permalink:: http://dlmf.nist.gov/3.4.E34
Encodings:: TeX, pMML, png

The results in this subsection for the partial derivatives follow from Panow (1955, Table 10). Those for the Laplacian and the biharmonic operator follow from the formulas for the partial derivatives.

For additional formulas involving values of $\nabla^{2}u$ and $\nabla^{4}u$ on square, triangular, and cubic grids, see Collatz (1960, Table VI, pp. 542–546).