Numerical Methods - 3.3 Interpolation

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DLMF Formula Constraints Maple Mathematica Symbolic
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Mathematica 		Numeric
Maple 		Numeric
Mathematica
3.3#Ex3 n 0 = - 1 2 ⁒ ( n - Οƒ ) subscript 𝑛 0 1 2 𝑛 𝜎 {\displaystyle{\displaystyle n_{0}=-\tfrac{1}{2}(n-\sigma)}}
n_{0} = -\tfrac{1}{2}(n-\sigma)		 

n[0] = -(1)/(2)*(n - sigma)
 
Subscript[n, 0] == -Divide[1,2]*(n - \[Sigma])
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3.3#Ex4 n 1 = 1 2 ⁒ ( n + Οƒ ) subscript 𝑛 1 1 2 𝑛 𝜎 {\displaystyle{\displaystyle n_{1}=\tfrac{1}{2}(n+\sigma)}}
n_{1} = \tfrac{1}{2}(n+\sigma)		 

n[1] = (1)/(2)*(n + sigma)
 
Subscript[n, 1] == Divide[1,2]*(n + \[Sigma])
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3.3.E9 Οƒ = 1 2 ⁒ ( 1 - ( - 1 ) n ) 𝜎 1 2 1 superscript 1 𝑛 {\displaystyle{\displaystyle\sigma=\tfrac{1}{2}(1-(-1)^{n})}}
\sigma = \tfrac{1}{2}(1-(-1)^{n})		 

sigma = (1)/(2)*(1 -(- 1)^(n))
 
\[Sigma] == Divide[1,2]*(1 -(- 1)^(n))
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3.3.E14 f t = ( 1 - t ) ⁒ f 0 + t ⁒ f 1 + R 1 , t subscript 𝑓 𝑑 1 𝑑 subscript 𝑓 0 𝑑 subscript 𝑓 1 subscript 𝑅 1 𝑑 {\displaystyle{\displaystyle f_{t}=(1-t)f_{0}+tf_{1}+R_{1,t}}}
f_{t} = (1-t)f_{0}+tf_{1}+R_{1,t}		 0 < t , t < 1 formulae-sequence 0 𝑑 𝑑 1 {\displaystyle{\displaystyle 0<t,t<1}} 
f[t] = (1 - t)*f[0]+ t*f[1]+ R[1 , t]
 
Subscript[f, t] == (1 - t)*Subscript[f, 0]+ t*Subscript[f, 1]+ Subscript[R, 1 , t]
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3.3.E15 c 1 = 1 8 subscript 𝑐 1 1 8 {\displaystyle{\displaystyle c_{1}=\tfrac{1}{8}}}
c_{1} = \tfrac{1}{8}		 0 < t , t < 1 formulae-sequence 0 𝑑 𝑑 1 {\displaystyle{\displaystyle 0<t,t<1}} 
c[1] = (1)/(8)
 
Subscript[c, 1] == Divide[1,8]
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3.3.E16 f t = βˆ‘ k = - 1 1 A k 2 ⁒ f k + R 2 , t subscript 𝑓 𝑑 superscript subscript π‘˜ 1 1 superscript subscript 𝐴 π‘˜ 2 subscript 𝑓 π‘˜ subscript 𝑅 2 𝑑 {\displaystyle{\displaystyle f_{t}=\sum_{k=-1}^{1}A_{k}^{2}f_{k}+R_{2,t}}}
f_{t} = \sum_{k=-1}^{1}A_{k}^{2}f_{k}+R_{2,t}		 | t | < 1 𝑑 1 {\displaystyle{\displaystyle\left|t\right|<1}} 
f[t] = sum((A[k])^(2)*f[k], k = - 1..1)+ R[2 , t]
 
Subscript[f, t] == Sum[(Subscript[A, k])^(2)*Subscript[f, k], {k, - 1, 1}, GenerateConditions->None]+ Subscript[R, 2 , t]
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3.3#Ex5 A - 1 2 = 1 2 ⁒ t ⁒ ( t - 1 ) superscript subscript 𝐴 1 2 1 2 𝑑 𝑑 1 {\displaystyle{\displaystyle A_{-1}^{2}=\tfrac{1}{2}t(t-1)}}
A_{-1}^{2} = \tfrac{1}{2}t(t-1)		 

(A[- 1])^(2) = (1)/(2)*t*(t - 1)
 
(Subscript[A, - 1])^(2) == Divide[1,2]*t*(t - 1)
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3.3#Ex6 A 0 2 = 1 - t 2 superscript subscript 𝐴 0 2 1 superscript 𝑑 2 {\displaystyle{\displaystyle A_{0}^{2}=1-t^{2}}}
A_{0}^{2} = 1-t^{2}		 

(A[0])^(2) = 1 - (t)^(2)
 
(Subscript[A, 0])^(2) == 1 - (t)^(2)
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3.3#Ex7 A 1 2 = 1 2 ⁒ t ⁒ ( t + 1 ) superscript subscript 𝐴 1 2 1 2 𝑑 𝑑 1 {\displaystyle{\displaystyle A_{1}^{2}=\tfrac{1}{2}t(t+1)}}
A_{1}^{2} = \tfrac{1}{2}t(t+1)		 

(A[1])^(2) = (1)/(2)*t*(t + 1)
 
(Subscript[A, 1])^(2) == Divide[1,2]*t*(t + 1)
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3.3.E18 c 2 = 1 / ( 9 ⁒ 3 ) subscript 𝑐 2 1 9 3 {\displaystyle{\displaystyle c_{2}=1/(9\sqrt{3})}}
c_{2} = 1/(9\sqrt{3})		 | t | < 1 𝑑 1 {\displaystyle{\displaystyle\left|t\right|<1}} 
c[2] = 1/(9*sqrt(3))
 
Subscript[c, 2] == 1/(9*Sqrt[3])
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3.3.E19 f t = βˆ‘ k = - 1 2 A k 3 ⁒ f k + R 3 , t subscript 𝑓 𝑑 superscript subscript π‘˜ 1 2 superscript subscript 𝐴 π‘˜ 3 subscript 𝑓 π‘˜ subscript 𝑅 3 𝑑 {\displaystyle{\displaystyle f_{t}=\sum_{k=-1}^{2}A_{k}^{3}f_{k}+R_{3,t}}}
f_{t} = \sum_{k=-1}^{2}A_{k}^{3}f_{k}+R_{3,t}		 - 1 < t , t < 2 formulae-sequence 1 𝑑 𝑑 2 {\displaystyle{\displaystyle-1<t,t<2}} 
f[t] = sum((A[k])^(3)*f[k], k = - 1..2)+ R[3 , t]
 
Subscript[f, t] == Sum[(Subscript[A, k])^(3)*Subscript[f, k], {k, - 1, 2}, GenerateConditions->None]+ Subscript[R, 3 , t]
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3.3#Ex8 A - 1 3 = - 1 6 ⁒ t ⁒ ( t - 1 ) ⁒ ( t - 2 ) superscript subscript 𝐴 1 3 1 6 𝑑 𝑑 1 𝑑 2 {\displaystyle{\displaystyle A_{-1}^{3}=-\tfrac{1}{6}t(t-1)(t-2)}}
A_{-1}^{3} = -\tfrac{1}{6}t(t-1)(t-2)		 

(A[- 1])^(3) = -(1)/(6)*t*(t - 1)*(t - 2)
 
(Subscript[A, - 1])^(3) == -Divide[1,6]*t*(t - 1)*(t - 2)
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3.3#Ex9 A 0 3 = 1 2 ⁒ ( t 2 - 1 ) ⁒ ( t - 2 ) superscript subscript 𝐴 0 3 1 2 superscript 𝑑 2 1 𝑑 2 {\displaystyle{\displaystyle A_{0}^{3}=\tfrac{1}{2}(t^{2}-1)(t-2)}}
A_{0}^{3} = \tfrac{1}{2}(t^{2}-1)(t-2)		 

(A[0])^(3) = (1)/(2)*((t)^(2)- 1)*(t - 2)
 
(Subscript[A, 0])^(3) == Divide[1,2]*((t)^(2)- 1)*(t - 2)
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3.3#Ex10 A 1 3 = - 1 2 ⁒ t ⁒ ( t + 1 ) ⁒ ( t - 2 ) superscript subscript 𝐴 1 3 1 2 𝑑 𝑑 1 𝑑 2 {\displaystyle{\displaystyle A_{1}^{3}=-\tfrac{1}{2}t(t+1)(t-2)}}
A_{1}^{3} = -\tfrac{1}{2}t(t+1)(t-2)		 

(A[1])^(3) = -(1)/(2)*t*(t + 1)*(t - 2)
 
(Subscript[A, 1])^(3) == -Divide[1,2]*t*(t + 1)*(t - 2)
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3.3#Ex11 A 2 3 = 1 6 ⁒ t ⁒ ( t 2 - 1 ) superscript subscript 𝐴 2 3 1 6 𝑑 superscript 𝑑 2 1 {\displaystyle{\displaystyle A_{2}^{3}=\tfrac{1}{6}t(t^{2}-1)}}
A_{2}^{3} = \tfrac{1}{6}t(t^{2}-1)		 

(A[2])^(3) = (1)/(6)*t*((t)^(2)- 1)
 
(Subscript[A, 2])^(3) == Divide[1,6]*t*((t)^(2)- 1)
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3.3.E22 f t = βˆ‘ k = - 2 2 A k 4 ⁒ f k + R 4 , t subscript 𝑓 𝑑 superscript subscript π‘˜ 2 2 superscript subscript 𝐴 π‘˜ 4 subscript 𝑓 π‘˜ subscript 𝑅 4 𝑑 {\displaystyle{\displaystyle f_{t}=\sum_{k=-2}^{2}A_{k}^{4}f_{k}+R_{4,t}}}
f_{t} = \sum_{k=-2}^{2}A_{k}^{4}f_{k}+R_{4,t}		 | t | < 2 𝑑 2 {\displaystyle{\displaystyle\left|t\right|<2}} 
f[t] = sum((A[k])^(4)*f[k], k = - 2..2)+ R[4 , t]
 
Subscript[f, t] == Sum[(Subscript[A, k])^(4)*Subscript[f, k], {k, - 2, 2}, GenerateConditions->None]+ Subscript[R, 4 , t]
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3.3#Ex12 A - 2 4 = 1 24 ⁒ t ⁒ ( t 2 - 1 ) ⁒ ( t - 2 ) superscript subscript 𝐴 2 4 1 24 𝑑 superscript 𝑑 2 1 𝑑 2 {\displaystyle{\displaystyle A_{-2}^{4}=\tfrac{1}{24}t(t^{2}-1)(t-2)}}
A_{-2}^{4} = \tfrac{1}{24}t(t^{2}-1)(t-2)		 

(A[- 2])^(4) = (1)/(24)*t*((t)^(2)- 1)*(t - 2)
 
(Subscript[A, - 2])^(4) == Divide[1,24]*t*((t)^(2)- 1)*(t - 2)
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3.3#Ex13 A - 1 4 = - 1 6 ⁒ t ⁒ ( t - 1 ) ⁒ ( t 2 - 4 ) superscript subscript 𝐴 1 4 1 6 𝑑 𝑑 1 superscript 𝑑 2 4 {\displaystyle{\displaystyle A_{-1}^{4}=-\tfrac{1}{6}t(t-1)(t^{2}-4)}}
A_{-1}^{4} = -\tfrac{1}{6}t(t-1)(t^{2}-4)		 

(A[- 1])^(4) = -(1)/(6)*t*(t - 1)*((t)^(2)- 4)
 
(Subscript[A, - 1])^(4) == -Divide[1,6]*t*(t - 1)*((t)^(2)- 4)
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3.3#Ex14 A 0 4 = 1 4 ⁒ ( t 2 - 1 ) ⁒ ( t 2 - 4 ) superscript subscript 𝐴 0 4 1 4 superscript 𝑑 2 1 superscript 𝑑 2 4 {\displaystyle{\displaystyle A_{0}^{4}=\tfrac{1}{4}(t^{2}-1)(t^{2}-4)}}
A_{0}^{4} = \tfrac{1}{4}(t^{2}-1)(t^{2}-4)		 

(A[0])^(4) = (1)/(4)*((t)^(2)- 1)*((t)^(2)- 4)
 
(Subscript[A, 0])^(4) == Divide[1,4]*((t)^(2)- 1)*((t)^(2)- 4)
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3.3#Ex15 A 1 4 = - 1 6 ⁒ t ⁒ ( t + 1 ) ⁒ ( t 2 - 4 ) superscript subscript 𝐴 1 4 1 6 𝑑 𝑑 1 superscript 𝑑 2 4 {\displaystyle{\displaystyle A_{1}^{4}=-\tfrac{1}{6}t(t+1)(t^{2}-4)}}
A_{1}^{4} = -\tfrac{1}{6}t(t+1)(t^{2}-4)		 

(A[1])^(4) = -(1)/(6)*t*(t + 1)*((t)^(2)- 4)
 
(Subscript[A, 1])^(4) == -Divide[1,6]*t*(t + 1)*((t)^(2)- 4)
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3.3#Ex16 A 2 4 = 1 24 ⁒ t ⁒ ( t 2 - 1 ) ⁒ ( t + 2 ) superscript subscript 𝐴 2 4 1 24 𝑑 superscript 𝑑 2 1 𝑑 2 {\displaystyle{\displaystyle A_{2}^{4}=\tfrac{1}{24}t(t^{2}-1)(t+2)}}
A_{2}^{4} = \tfrac{1}{24}t(t^{2}-1)(t+2)		 

(A[2])^(4) = (1)/(24)*t*((t)^(2)- 1)*(t + 2)
 
(Subscript[A, 2])^(4) == Divide[1,24]*t*((t)^(2)- 1)*(t + 2)
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3.3.E25 f t = βˆ‘ k = - 2 3 A k 5 ⁒ f k + R 5 , t subscript 𝑓 𝑑 superscript subscript π‘˜ 2 3 superscript subscript 𝐴 π‘˜ 5 subscript 𝑓 π‘˜ subscript 𝑅 5 𝑑 {\displaystyle{\displaystyle f_{t}=\sum_{k=-2}^{3}A_{k}^{5}f_{k}+R_{5,t}}}
f_{t} = \sum_{k=-2}^{3}A_{k}^{5}f_{k}+R_{5,t}		 - 2 < t , t < 3 formulae-sequence 2 𝑑 𝑑 3 {\displaystyle{\displaystyle-2<t,t<3}} 
f[t] = sum((A[k])^(5)*f[k], k = - 2..3)+ R[5 , t]
 
Subscript[f, t] == Sum[(Subscript[A, k])^(5)*Subscript[f, k], {k, - 2, 3}, GenerateConditions->None]+ Subscript[R, 5 , t]
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3.3#Ex17 A - 2 5 = - 1 120 ⁒ t ⁒ ( t 2 - 1 ) ⁒ ( t - 2 ) ⁒ ( t - 3 ) superscript subscript 𝐴 2 5 1 120 𝑑 superscript 𝑑 2 1 𝑑 2 𝑑 3 {\displaystyle{\displaystyle A_{-2}^{5}=-\tfrac{1}{120}t(t^{2}-1)(t-2)(t-3)}}
A_{-2}^{5} = -\tfrac{1}{120}t(t^{2}-1)(t-2)(t-3)		 

(A[- 2])^(5) = -(1)/(120)*t*((t)^(2)- 1)*(t - 2)*(t - 3)
 
(Subscript[A, - 2])^(5) == -Divide[1,120]*t*((t)^(2)- 1)*(t - 2)*(t - 3)
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3.3#Ex18 A - 1 5 = 1 24 ⁒ t ⁒ ( t - 1 ) ⁒ ( t 2 - 4 ) ⁒ ( t - 3 ) superscript subscript 𝐴 1 5 1 24 𝑑 𝑑 1 superscript 𝑑 2 4 𝑑 3 {\displaystyle{\displaystyle A_{-1}^{5}=\tfrac{1}{24}t(t-1)(t^{2}-4)(t-3)}}
A_{-1}^{5} = \tfrac{1}{24}t(t-1)(t^{2}-4)(t-3)		 

(A[- 1])^(5) = (1)/(24)*t*(t - 1)*((t)^(2)- 4)*(t - 3)
 
(Subscript[A, - 1])^(5) == Divide[1,24]*t*(t - 1)*((t)^(2)- 4)*(t - 3)
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3.3#Ex19 A 0 5 = - 1 12 ⁒ ( t 2 - 1 ) ⁒ ( t 2 - 4 ) ⁒ ( t - 3 ) superscript subscript 𝐴 0 5 1 12 superscript 𝑑 2 1 superscript 𝑑 2 4 𝑑 3 {\displaystyle{\displaystyle A_{0}^{5}=-\tfrac{1}{12}(t^{2}-1)(t^{2}-4)(t-3)}}
A_{0}^{5} = -\tfrac{1}{12}(t^{2}-1)(t^{2}-4)(t-3)		 

(A[0])^(5) = -(1)/(12)*((t)^(2)- 1)*((t)^(2)- 4)*(t - 3)
 
(Subscript[A, 0])^(5) == -Divide[1,12]*((t)^(2)- 1)*((t)^(2)- 4)*(t - 3)
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3.3#Ex20 A 1 5 = 1 12 ⁒ t ⁒ ( t + 1 ) ⁒ ( t 2 - 4 ) ⁒ ( t - 3 ) superscript subscript 𝐴 1 5 1 12 𝑑 𝑑 1 superscript 𝑑 2 4 𝑑 3 {\displaystyle{\displaystyle A_{1}^{5}=\tfrac{1}{12}t(t+1)(t^{2}-4)(t-3)}}
A_{1}^{5} = \tfrac{1}{12}t(t+1)(t^{2}-4)(t-3)		 

(A[1])^(5) = (1)/(12)*t*(t + 1)*((t)^(2)- 4)*(t - 3)
 
(Subscript[A, 1])^(5) == Divide[1,12]*t*(t + 1)*((t)^(2)- 4)*(t - 3)
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3.3#Ex21 A 2 5 = - 1 24 ⁒ t ⁒ ( t 2 - 1 ) ⁒ ( t + 2 ) ⁒ ( t - 3 ) superscript subscript 𝐴 2 5 1 24 𝑑 superscript 𝑑 2 1 𝑑 2 𝑑 3 {\displaystyle{\displaystyle A_{2}^{5}=-\tfrac{1}{24}t(t^{2}-1)(t+2)(t-3)}}
A_{2}^{5} = -\tfrac{1}{24}t(t^{2}-1)(t+2)(t-3)		 

(A[2])^(5) = -(1)/(24)*t*((t)^(2)- 1)*(t + 2)*(t - 3)
 
(Subscript[A, 2])^(5) == -Divide[1,24]*t*((t)^(2)- 1)*(t + 2)*(t - 3)
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3.3#Ex22 A 3 5 = 1 120 ⁒ t ⁒ ( t 2 - 1 ) ⁒ ( t 2 - 4 ) superscript subscript 𝐴 3 5 1 120 𝑑 superscript 𝑑 2 1 superscript 𝑑 2 4 {\displaystyle{\displaystyle A_{3}^{5}=\tfrac{1}{120}t(t^{2}-1)(t^{2}-4)}}
A_{3}^{5} = \tfrac{1}{120}t(t^{2}-1)(t^{2}-4)		 

(A[3])^(5) = (1)/(120)*t*((t)^(2)- 1)*((t)^(2)- 4)
 
(Subscript[A, 3])^(5) == Divide[1,120]*t*((t)^(2)- 1)*((t)^(2)- 4)
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3.3.E28 f t = βˆ‘ k = - 3 3 A k 6 ⁒ f k + R 6 , t subscript 𝑓 𝑑 superscript subscript π‘˜ 3 3 superscript subscript 𝐴 π‘˜ 6 subscript 𝑓 π‘˜ subscript 𝑅 6 𝑑 {\displaystyle{\displaystyle f_{t}=\sum_{k=-3}^{3}A_{k}^{6}f_{k}+R_{6,t}}}
f_{t} = \sum_{k=-3}^{3}A_{k}^{6}f_{k}+R_{6,t}		 | t | < 3 𝑑 3 {\displaystyle{\displaystyle\left|t\right|<3}} 
f[t] = sum((A[k])^(6)*f[k], k = - 3..3)+ R[6 , t]
 
Subscript[f, t] == Sum[(Subscript[A, k])^(6)*Subscript[f, k], {k, - 3, 3}, GenerateConditions->None]+ Subscript[R, 6 , t]
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3.3#Ex23 A - 3 6 = 1 720 ⁒ t ⁒ ( t 2 - 1 ) ⁒ ( t - 3 ) ⁒ ( t 2 - 4 ) superscript subscript 𝐴 3 6 1 720 𝑑 superscript 𝑑 2 1 𝑑 3 superscript 𝑑 2 4 {\displaystyle{\displaystyle A_{-3}^{6}=\tfrac{1}{720}t(t^{2}-1)(t-3)(t^{2}-4)}}
A_{-3}^{6} = \tfrac{1}{720}t(t^{2}-1)(t-3)(t^{2}-4)		 

(A[- 3])^(6) = (1)/(720)*t*((t)^(2)- 1)*(t - 3)*((t)^(2)- 4)
 
(Subscript[A, - 3])^(6) == Divide[1,720]*t*((t)^(2)- 1)*(t - 3)*((t)^(2)- 4)
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3.3#Ex24 A - 2 6 = - 1 120 ⁒ t ⁒ ( t 2 - 1 ) ⁒ ( t - 2 ) ⁒ ( t 2 - 9 ) superscript subscript 𝐴 2 6 1 120 𝑑 superscript 𝑑 2 1 𝑑 2 superscript 𝑑 2 9 {\displaystyle{\displaystyle A_{-2}^{6}=-\tfrac{1}{120}t(t^{2}-1)(t-2)(t^{2}-9% )}}
A_{-2}^{6} = -\tfrac{1}{120}t(t^{2}-1)(t-2)(t^{2}-9)		 

(A[- 2])^(6) = -(1)/(120)*t*((t)^(2)- 1)*(t - 2)*((t)^(2)- 9)
 
(Subscript[A, - 2])^(6) == -Divide[1,120]*t*((t)^(2)- 1)*(t - 2)*((t)^(2)- 9)
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3.3#Ex25 A - 1 6 = 1 48 ⁒ t ⁒ ( t - 1 ) ⁒ ( t 2 - 4 ) ⁒ ( t 2 - 9 ) superscript subscript 𝐴 1 6 1 48 𝑑 𝑑 1 superscript 𝑑 2 4 superscript 𝑑 2 9 {\displaystyle{\displaystyle A_{-1}^{6}=\tfrac{1}{48}t(t-1)(t^{2}-4)(t^{2}-9)}}
A_{-1}^{6} = \tfrac{1}{48}t(t-1)(t^{2}-4)(t^{2}-9)		 

(A[- 1])^(6) = (1)/(48)*t*(t - 1)*((t)^(2)- 4)*((t)^(2)- 9)
 
(Subscript[A, - 1])^(6) == Divide[1,48]*t*(t - 1)*((t)^(2)- 4)*((t)^(2)- 9)
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3.3#Ex26 A 0 6 = - 1 36 ⁒ ( t 2 - 1 ) ⁒ ( t 2 - 4 ) ⁒ ( t 2 - 9 ) superscript subscript 𝐴 0 6 1 36 superscript 𝑑 2 1 superscript 𝑑 2 4 superscript 𝑑 2 9 {\displaystyle{\displaystyle A_{0}^{6}=-\tfrac{1}{36}(t^{2}-1)(t^{2}-4)(t^{2}-% 9)}}
A_{0}^{6} = -\tfrac{1}{36}(t^{2}-1)(t^{2}-4)(t^{2}-9)		 

(A[0])^(6) = -(1)/(36)*((t)^(2)- 1)*((t)^(2)- 4)*((t)^(2)- 9)
 
(Subscript[A, 0])^(6) == -Divide[1,36]*((t)^(2)- 1)*((t)^(2)- 4)*((t)^(2)- 9)
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3.3#Ex27 A 1 6 = 1 48 ⁒ t ⁒ ( t + 1 ) ⁒ ( t 2 - 4 ) ⁒ ( t 2 - 9 ) superscript subscript 𝐴 1 6 1 48 𝑑 𝑑 1 superscript 𝑑 2 4 superscript 𝑑 2 9 {\displaystyle{\displaystyle A_{1}^{6}=\tfrac{1}{48}t(t+1)(t^{2}-4)(t^{2}-9)}}
A_{1}^{6} = \tfrac{1}{48}t(t+1)(t^{2}-4)(t^{2}-9)		 

(A[1])^(6) = (1)/(48)*t*(t + 1)*((t)^(2)- 4)*((t)^(2)- 9)
 
(Subscript[A, 1])^(6) == Divide[1,48]*t*(t + 1)*((t)^(2)- 4)*((t)^(2)- 9)
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3.3#Ex28 A 2 6 = - 1 120 ⁒ t ⁒ ( t 2 - 1 ) ⁒ ( t + 2 ) ⁒ ( t 2 - 9 ) superscript subscript 𝐴 2 6 1 120 𝑑 superscript 𝑑 2 1 𝑑 2 superscript 𝑑 2 9 {\displaystyle{\displaystyle A_{2}^{6}=-\tfrac{1}{120}t(t^{2}-1)(t+2)(t^{2}-9)}}
A_{2}^{6} = -\tfrac{1}{120}t(t^{2}-1)(t+2)(t^{2}-9)		 

(A[2])^(6) = -(1)/(120)*t*((t)^(2)- 1)*(t + 2)*((t)^(2)- 9)
 
(Subscript[A, 2])^(6) == -Divide[1,120]*t*((t)^(2)- 1)*(t + 2)*((t)^(2)- 9)
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3.3#Ex29 A 3 6 = 1 720 ⁒ t ⁒ ( t 2 - 1 ) ⁒ ( t + 3 ) ⁒ ( t 2 - 4 ) superscript subscript 𝐴 3 6 1 720 𝑑 superscript 𝑑 2 1 𝑑 3 superscript 𝑑 2 4 {\displaystyle{\displaystyle A_{3}^{6}=\tfrac{1}{720}t(t^{2}-1)(t+3)(t^{2}-4)}}
A_{3}^{6} = \tfrac{1}{720}t(t^{2}-1)(t+3)(t^{2}-4)		 

(A[3])^(6) = (1)/(720)*t*((t)^(2)- 1)*(t + 3)*((t)^(2)- 4)
 
(Subscript[A, 3])^(6) == Divide[1,720]*t*((t)^(2)- 1)*(t + 3)*((t)^(2)- 4)
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3.3.E31 f t = βˆ‘ k = - 3 4 A k 7 ⁒ f k + R 7 , t subscript 𝑓 𝑑 superscript subscript π‘˜ 3 4 superscript subscript 𝐴 π‘˜ 7 subscript 𝑓 π‘˜ subscript 𝑅 7 𝑑 {\displaystyle{\displaystyle f_{t}=\sum_{k=-3}^{4}A_{k}^{7}f_{k}+R_{7,t}}}
f_{t} = \sum_{k=-3}^{4}A_{k}^{7}f_{k}+R_{7,t}		 - 3 < t , t < 4 formulae-sequence 3 𝑑 𝑑 4 {\displaystyle{\displaystyle-3<t,t<4}} 
f[t] = sum((A[k])^(7)*f[k], k = - 3..4)+ R[7 , t]
 
Subscript[f, t] == Sum[(Subscript[A, k])^(7)*Subscript[f, k], {k, - 3, 4}, GenerateConditions->None]+ Subscript[R, 7 , t]
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3.3#Ex30 A - 3 7 = - 1 5040 ⁒ t ⁒ ( t 2 - 1 ) ⁒ ( t - 3 ) ⁒ ( t - 4 ) ⁒ ( t 2 - 4 ) superscript subscript 𝐴 3 7 1 5040 𝑑 superscript 𝑑 2 1 𝑑 3 𝑑 4 superscript 𝑑 2 4 {\displaystyle{\displaystyle A_{-3}^{7}=-\tfrac{1}{5040}t(t^{2}-1)(t-3)(t-4)(t% ^{2}-4)}}
A_{-3}^{7} = -\tfrac{1}{5040}t(t^{2}-1)(t-3)(t-4)(t^{2}-4)		 

(A[- 3])^(7) = -(1)/(5040)*t*((t)^(2)- 1)*(t - 3)*(t - 4)*((t)^(2)- 4)
 
(Subscript[A, - 3])^(7) == -Divide[1,5040]*t*((t)^(2)- 1)*(t - 3)*(t - 4)*((t)^(2)- 4)
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3.3#Ex31 A - 2 7 = 1 720 ⁒ t ⁒ ( t 2 - 1 ) ⁒ ( t - 2 ) ⁒ ( t - 4 ) ⁒ ( t 2 - 9 ) superscript subscript 𝐴 2 7 1 720 𝑑 superscript 𝑑 2 1 𝑑 2 𝑑 4 superscript 𝑑 2 9 {\displaystyle{\displaystyle A_{-2}^{7}=\tfrac{1}{720}t(t^{2}-1)(t-2)(t-4)(t^{% 2}-9)}}
A_{-2}^{7} = \tfrac{1}{720}t(t^{2}-1)(t-2)(t-4)(t^{2}-9)		 

(A[- 2])^(7) = (1)/(720)*t*((t)^(2)- 1)*(t - 2)*(t - 4)*((t)^(2)- 9)
 
(Subscript[A, - 2])^(7) == Divide[1,720]*t*((t)^(2)- 1)*(t - 2)*(t - 4)*((t)^(2)- 9)
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3.3#Ex32 A - 1 7 = - 1 240 ⁒ t ⁒ ( t - 1 ) ⁒ ( t - 4 ) ⁒ ( t 2 - 4 ) ⁒ ( t 2 - 9 ) superscript subscript 𝐴 1 7 1 240 𝑑 𝑑 1 𝑑 4 superscript 𝑑 2 4 superscript 𝑑 2 9 {\displaystyle{\displaystyle A_{-1}^{7}=-\tfrac{1}{240}t(t-1)(t-4)(t^{2}-4)(t^% {2}-9)}}
A_{-1}^{7} = -\tfrac{1}{240}t(t-1)(t-4)(t^{2}-4)(t^{2}-9)		 

(A[- 1])^(7) = -(1)/(240)*t*(t - 1)*(t - 4)*((t)^(2)- 4)*((t)^(2)- 9)
 
(Subscript[A, - 1])^(7) == -Divide[1,240]*t*(t - 1)*(t - 4)*((t)^(2)- 4)*((t)^(2)- 9)
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3.3#Ex33 A 0 7 = 1 144 ⁒ ( t 2 - 1 ) ⁒ ( t - 4 ) ⁒ ( t 2 - 4 ) ⁒ ( t 2 - 9 ) superscript subscript 𝐴 0 7 1 144 superscript 𝑑 2 1 𝑑 4 superscript 𝑑 2 4 superscript 𝑑 2 9 {\displaystyle{\displaystyle A_{0}^{7}=\tfrac{1}{144}(t^{2}-1)(t-4)(t^{2}-4)(t% ^{2}-9)}}
A_{0}^{7} = \tfrac{1}{144}(t^{2}-1)(t-4)(t^{2}-4)(t^{2}-9)		 

(A[0])^(7) = (1)/(144)*((t)^(2)- 1)*(t - 4)*((t)^(2)- 4)*((t)^(2)- 9)
 
(Subscript[A, 0])^(7) == Divide[1,144]*((t)^(2)- 1)*(t - 4)*((t)^(2)- 4)*((t)^(2)- 9)
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3.3#Ex34 A 1 7 = - 1 144 ⁒ t ⁒ ( t + 1 ) ⁒ ( t - 4 ) ⁒ ( t 2 - 4 ) ⁒ ( t 2 - 9 ) superscript subscript 𝐴 1 7 1 144 𝑑 𝑑 1 𝑑 4 superscript 𝑑 2 4 superscript 𝑑 2 9 {\displaystyle{\displaystyle A_{1}^{7}=-\tfrac{1}{144}t(t+1)(t-4)(t^{2}-4)(t^{% 2}-9)}}
A_{1}^{7} = -\tfrac{1}{144}t(t+1)(t-4)(t^{2}-4)(t^{2}-9)		 

(A[1])^(7) = -(1)/(144)*t*(t + 1)*(t - 4)*((t)^(2)- 4)*((t)^(2)- 9)
 
(Subscript[A, 1])^(7) == -Divide[1,144]*t*(t + 1)*(t - 4)*((t)^(2)- 4)*((t)^(2)- 9)
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3.3#Ex35 A 2 7 = 1 240 ⁒ t ⁒ ( t 2 - 1 ) ⁒ ( t + 2 ) ⁒ ( t - 4 ) ⁒ ( t 2 - 9 ) superscript subscript 𝐴 2 7 1 240 𝑑 superscript 𝑑 2 1 𝑑 2 𝑑 4 superscript 𝑑 2 9 {\displaystyle{\displaystyle A_{2}^{7}=\tfrac{1}{240}t(t^{2}-1)(t+2)(t-4)(t^{2% }-9)}}
A_{2}^{7} = \tfrac{1}{240}t(t^{2}-1)(t+2)(t-4)(t^{2}-9)		 

(A[2])^(7) = (1)/(240)*t*((t)^(2)- 1)*(t + 2)*(t - 4)*((t)^(2)- 9)
 
(Subscript[A, 2])^(7) == Divide[1,240]*t*((t)^(2)- 1)*(t + 2)*(t - 4)*((t)^(2)- 9)
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3.3#Ex36 A 3 7 = - 1 720 ⁒ t ⁒ ( t 2 - 1 ) ⁒ ( t + 3 ) ⁒ ( t - 4 ) ⁒ ( t 2 - 4 ) superscript subscript 𝐴 3 7 1 720 𝑑 superscript 𝑑 2 1 𝑑 3 𝑑 4 superscript 𝑑 2 4 {\displaystyle{\displaystyle A_{3}^{7}=-\tfrac{1}{720}t(t^{2}-1)(t+3)(t-4)(t^{% 2}-4)}}
A_{3}^{7} = -\tfrac{1}{720}t(t^{2}-1)(t+3)(t-4)(t^{2}-4)		 

(A[3])^(7) = -(1)/(720)*t*((t)^(2)- 1)*(t + 3)*(t - 4)*((t)^(2)- 4)
 
(Subscript[A, 3])^(7) == -Divide[1,720]*t*((t)^(2)- 1)*(t + 3)*(t - 4)*((t)^(2)- 4)
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3.3#Ex37 A 4 7 = 1 5040 ⁒ t ⁒ ( t 2 - 1 ) ⁒ ( t 2 - 4 ) ⁒ ( t 2 - 9 ) superscript subscript 𝐴 4 7 1 5040 𝑑 superscript 𝑑 2 1 superscript 𝑑 2 4 superscript 𝑑 2 9 {\displaystyle{\displaystyle A_{4}^{7}=\tfrac{1}{5040}t(t^{2}-1)(t^{2}-4)(t^{2% }-9)}}
A_{4}^{7} = \tfrac{1}{5040}t(t^{2}-1)(t^{2}-4)(t^{2}-9)		 

(A[4])^(7) = (1)/(5040)*t*((t)^(2)- 1)*((t)^(2)- 4)*((t)^(2)- 9)
 
(Subscript[A, 4])^(7) == Divide[1,5040]*t*((t)^(2)- 1)*((t)^(2)- 4)*((t)^(2)- 9)
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3.3#Ex38 f = f 0 𝑓 subscript 𝑓 0 {\displaystyle{\displaystyle f=f_{0}}}
f = f_{0}		 

f = f[0]
 
f == Subscript[f, 0]
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3.3#Ex39 f = ( [ z 1 ] ⁒ f - [ z 0 ] ⁒ f ) / ( z 1 - z 0 ) 𝑓 delimited-[] subscript 𝑧 1 𝑓 delimited-[] subscript 𝑧 0 𝑓 subscript 𝑧 1 subscript 𝑧 0 {\displaystyle{\displaystyle f=({[z_{1}]}f-{[z_{0}]}f)/(z_{1}-z_{0})}}
f = ({[z_{1}]}f-{[z_{0}]}f)/(z_{1}-z_{0})		 

f = ((z[1])*f -(z[0])*f)/(z[1]- z[0])
 
f == ((Subscript[z, 1])*f -(Subscript[z, 0])*f)/(Subscript[z, 1]- Subscript[z, 0])
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3.3#Ex40 f = ( [ z 1 , z 2 ] ⁒ f - [ z 0 , z 1 ] ⁒ f ) / ( z 2 - z 0 ) 𝑓 subscript 𝑧 1 subscript 𝑧 2 𝑓 subscript 𝑧 0 subscript 𝑧 1 𝑓 subscript 𝑧 2 subscript 𝑧 0 {\displaystyle{\displaystyle f=({[z_{1},z_{2}]}f-{[z_{0},z_{1}]}f)/(z_{2}-z_{0% })}}
f = ({[z_{1},z_{2}]}f-{[z_{0},z_{1}]}f)/(z_{2}-z_{0})		 

f = ([z[1], z[2]]*f -[z[0], z[1]]*f)/(z[2]- z[0])
 
f == ([Subscript[z, 1], Subscript[z, 2]]*f -[Subscript[z, 0], Subscript[z, 1]]*f)/(Subscript[z, 2]- Subscript[z, 0])
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3.3.E39 x ⁒ ( f ) = [ f 0 ] ⁒ x + ( f - f 0 ) ⁒ [ f 0 , f 1 ] ⁒ x + ( f - f 0 ) ⁒ ( f - f 1 ) ⁒ [ f 0 , f 1 , f 2 ] ⁒ x π‘₯ 𝑓 delimited-[] subscript 𝑓 0 π‘₯ 𝑓 subscript 𝑓 0 subscript 𝑓 0 subscript 𝑓 1 π‘₯ 𝑓 subscript 𝑓 0 𝑓 subscript 𝑓 1 subscript 𝑓 0 subscript 𝑓 1 subscript 𝑓 2 π‘₯ {\displaystyle{\displaystyle x(f)=[f_{0}]x+(f-f_{0})[f_{0},f_{1}]x+(f-f_{0})(f% -f_{1})[f_{0},f_{1},f_{2}]x}}
x(f) = [f_{0}]x+(f-f_{0})[f_{0},f_{1}]x+(f-f_{0})(f-f_{1})[f_{0},f_{1},f_{2}]x		 

x(f) = (f[0])*x(+)*(f - f[0])*[f[0], f[1]]*x(+)*(f - f[0])*(f - f[1])*[f[0], f[1], f[2]]*x
 
x[f] == (Subscript[f, 0])*x[+]*(f - Subscript[f, 0])*[Subscript[f, 0], Subscript[f, 1]]*x[+]*(f - Subscript[f, 0])*(f - Subscript[f, 1])*[Subscript[f, 0], Subscript[f, 1], Subscript[f, 2]]*x
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3.3.E40 x = - 2.2 + 1.44011 1973 ⁒ ( f - 0.09614 53780 ) + 0.08865 85832 ⁒ ( f - 0.09614 53780 ) ⁒ ( f - 0.02670 63331 ) π‘₯ 2.2 1.44011 1973 𝑓 0.09614 53780 0.08865 85832 𝑓 0.09614 53780 𝑓 0.02670 63331 {\displaystyle{\displaystyle x=-2.2+1.44011\;1973(f-0.09614\;53780)+0.08865\;8% 5832\*(f-0.09614\;53780)(f-0.02670\;63331)}}
x = -2.2+1.44011\;1973(f-0.09614\;53780)+0.08865\;85832\*(f-0.09614\;53780)(f-0.02670\;63331)		 

x = - 2.2 + 1.440111973*(f - 0.0961453780)+ 0.0886585832 *(f - 0.0961453780)*(f - 0.0267063331)
 
x == - 2.2 + 1.440111973*(f - 0.0961453780)+ 0.0886585832 *(f - 0.0961453780)*(f - 0.0267063331)
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