นิพจน์ 13: "K" equals left bracket, 0...1 7 , right bracketK=0...17
equals=
00
11
22
33
44
55
66
77
88
99
1010
1111
1212
1313
1414
ลิสต์ 18 จำนวน
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นิพจน์ 14: "f" left parenthesis, "x" , right parenthesis equals hyperbolic secant left parenthesis, "x" , right parenthesis left parenthesis, 1 quarter plus 1 half sin left parenthesis, 3 pi tanh left parenthesis, StartFraction, "x" Over 4 , EndFraction , right parenthesis , right parenthesis , right parenthesisfx=sechx14+12sin3πtanhx4
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นิพจน์ 15: "C" Subscript, "T" , Baseline equals StartFraction, left parenthesis, Start integral from negative infinity to infinity , end integral, StartNestedFraction, "T" left parenthesis, "K" , "x" , right parenthesis "f" left parenthesis, "x" , right parenthesis NestedOver left parenthesis, 1 plus "x" squared , right parenthesis , EndNestedFraction "d" "x" , right parenthesis Over left parenthesis, Start integral from negative infinity to infinity , end integral, StartNestedFraction, "T" left parenthesis, "K" , "x" , right parenthesis squared NestedOver left parenthesis, 1 plus "x" squared , right parenthesis , EndNestedFraction "d" "x" , right parenthesis , EndFractionCT=∫∞−∞TK,xfx1+x2dx∫∞−∞TK,x21+x2dx
นิพจน์ 16: "C" Subscript, "U" , Baseline equals StartFraction, left parenthesis, Start integral from negative infinity to infinity , end integral, StartNestedFraction, "U" left parenthesis, "K" , "x" , right parenthesis "f" left parenthesis, "x" , right parenthesis NestedOver left parenthesis, 1 plus "x" squared , right parenthesis , EndNestedFraction "d" "x" , right parenthesis Over left parenthesis, Start integral from negative infinity to infinity , end integral, StartNestedFraction, "U" left parenthesis, "K" , "x" , right parenthesis squared NestedOver left parenthesis, 1 plus "x" squared , right parenthesis , EndNestedFraction "d" "x" , right parenthesis , EndFractionCU=∫∞−∞UK,xfx1+x2dx∫∞−∞UK,x21+x2dx
นิพจน์ 17: Start sum from "n" equals 0 to "N" , end sum, left parenthesis, "C" Subscript, "T" , Baseline left bracket, "n" plus 1 , right bracket "T" left parenthesis, "n" , "x" , right parenthesis , right parenthesisN∑n=0CTn+1Tn,x
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นิพจน์ 18: Start sum from "n" equals 0 to "N" , end sum, left parenthesis, "C" Subscript, "U" , Baseline left bracket, "n" plus 1 , right bracket "U" left parenthesis, "n" , "x" , right parenthesis , right parenthesisN∑n=0CUn+1Un,x
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You can tweak the weight function in the numerator and then apply the reciprocal of the modified version to change asymptotic behavior or tradeoff error in different intervals (so long as it doesn't break convergence)
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นิพจน์ 20: "C" Subscript, "T" 2 , Baseline equals StartFraction, left parenthesis, Start integral from negative infinity to infinity , end integral, StartNestedFraction, "T" left parenthesis, "K" , "x" , right parenthesis "f" left parenthesis, "x" , right parenthesis NestedOver left parenthesis, 1 plus "x" squared , right parenthesis Superscript, left parenthesis, 1 plus "a" , right parenthesis , Baseline , EndNestedFraction "d" "x" , right parenthesis Over left parenthesis, Start integral from negative infinity to infinity , end integral, StartNestedFraction, "T" left parenthesis, "K" , "x" , right parenthesis squared NestedOver left parenthesis, 1 plus "x" squared , right parenthesis , EndNestedFraction "d" "x" , right parenthesis , EndFractionCT2=∫∞−∞TK,xfx1+x21+adx∫∞−∞TK,x21+x2dx
นิพจน์ 21: "C" Subscript, "U" 2 , Baseline equals StartFraction, left parenthesis, Start integral from negative infinity to infinity , end integral, StartNestedFraction, "U" left parenthesis, "K" , "x" , right parenthesis "f" left parenthesis, "x" , right parenthesis NestedOver left parenthesis, 1 plus "x" squared , right parenthesis Superscript, left parenthesis, 1 plus "a" , right parenthesis , Baseline , EndNestedFraction "d" "x" , right parenthesis Over left parenthesis, Start integral from negative infinity to infinity , end integral, StartNestedFraction, "U" left parenthesis, "K" , "x" , right parenthesis squared NestedOver left parenthesis, 1 plus "x" squared , right parenthesis , EndNestedFraction "d" "x" , right parenthesis , EndFractionCU2=∫∞−∞UK,xfx1+x21+adx∫∞−∞UK,x21+x2dx
นิพจน์ 23: Start sum from "n" equals 0 to "N" , end sum, left parenthesis, "C" Subscript, "T" 2 , Baseline left bracket, "n" plus 1 , right bracket "T" left parenthesis, "n" , "x" , right parenthesis , right parenthesis left parenthesis, 1 plus "x" squared , right parenthesis Superscript, left parenthesis, "a" , right parenthesis , BaselineN∑n=0CT2n+1Tn,x1+x2a
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นิพจน์ 24: Start sum from "n" equals 0 to "N" , end sum, left parenthesis, "C" Subscript, "U" 2 , Baseline left bracket, "n" plus 1 , right bracket "U" left parenthesis, "n" , "x" , right parenthesis , right parenthesis left parenthesis, 1 plus "x" squared , right parenthesis Superscript, left parenthesis, "a" , right parenthesis , BaselineN∑n=0CU2n+1Un,x1+x2a
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Coefficients can also using a truncated fourier transform of f(tan(pi/2*x)) on (-1,1)