No matter how you are alone, no matter where you live in a marginal land, if you read one line per line firmly, you will surely have a moment to understand.
ˇCebyˇsev polynomials and the hyperbolic polynomials
3.1.1 Tk+1 = 2xTk−Tk−1
3.1.2 T1 = x, T2 = 2x2 −1
3.1.3 gk+1 = 2xgk−gk−1
3.1.4 g1 = 1, g2 = 2x
3.2 Kurokawa polynomials
3.2.1 Theorem
3.2.4 cos2nθ = Kn + 2Kn′sin2 θ
3.2.5 Kn = 1 n gn
3.2.6 Tn = xKn+ 1−x2 2 Kn′
3.2.7 nTn = xgn−gn′
3.2.8 nTn′ = gn + xgn′ + 2xgn′−gn′′
3.2.9 y + 3xy′−y′′ = 0
3.2.10 −ny + 3xy′−y′′ = 0
3.3 Method of solution of second order linear homogeneous differential equations by series
3.3.1 P0y′′ + P1y′ + P2y = 0
3.3.2 −ny + 3xy′−y′′ = 0
3.3.4 −n2y + xy′−y′′ = 0
3.3.6 Theorem(Solutions of the second order linear homogeneous differential equation −n2y + xy′−y′′ = 0)
3.3.7 −y + 8y′ + 4xy′′ = 0
3.4 New discussion of ˇCebyˇsev’s differential equation
3.5 New discussion of the hyperbolic differential equation
ˇCebyˇsev function
3.6.1 −ν2y + xy′−y′′ = 0
3.6.2 Theorem
3.7 First kind hyperbolic function
3.7.1 −y + 8y′ + 4xy′′ = 0
3.7.2 y = Wν = sin(ν arccos(/2)/2) sin(arccos(/2)/2)
3.8 Envelope
3.8.1 Definition
3.8.2 Theorem
3.8.3 fx(g,h,ν)g′ + fy(g,h,ν)h′ + fν(g,h,ν) = 0
3.8.4 fx(g,h,ν)g′ + fy(g,h,ν)h′ = 0
3.8.5 Remark
3.9 The envelope of curve group
3.10 The envelope of curve group
3.10.1 y2arccos2 = arccos2
3.11 The envelope of curve group
3.11.1 y2 sin2(arccos(/2)/2) = 1 or arccos2(/2) = 0
Language
English
Pages
45
Format
Kindle Edition
Release
October 17, 2017
Second order linear homogeneous differential equations which are satisfied by polynomials English edition
No matter how you are alone, no matter where you live in a marginal land, if you read one line per line firmly, you will surely have a moment to understand.
ˇCebyˇsev polynomials and the hyperbolic polynomials
3.1.1 Tk+1 = 2xTk−Tk−1
3.1.2 T1 = x, T2 = 2x2 −1
3.1.3 gk+1 = 2xgk−gk−1
3.1.4 g1 = 1, g2 = 2x
3.2 Kurokawa polynomials
3.2.1 Theorem
3.2.4 cos2nθ = Kn + 2Kn′sin2 θ
3.2.5 Kn = 1 n gn
3.2.6 Tn = xKn+ 1−x2 2 Kn′
3.2.7 nTn = xgn−gn′
3.2.8 nTn′ = gn + xgn′ + 2xgn′−gn′′
3.2.9 y + 3xy′−y′′ = 0
3.2.10 −ny + 3xy′−y′′ = 0
3.3 Method of solution of second order linear homogeneous differential equations by series
3.3.1 P0y′′ + P1y′ + P2y = 0
3.3.2 −ny + 3xy′−y′′ = 0
3.3.4 −n2y + xy′−y′′ = 0
3.3.6 Theorem(Solutions of the second order linear homogeneous differential equation −n2y + xy′−y′′ = 0)
3.3.7 −y + 8y′ + 4xy′′ = 0
3.4 New discussion of ˇCebyˇsev’s differential equation
3.5 New discussion of the hyperbolic differential equation
ˇCebyˇsev function
3.6.1 −ν2y + xy′−y′′ = 0
3.6.2 Theorem
3.7 First kind hyperbolic function
3.7.1 −y + 8y′ + 4xy′′ = 0
3.7.2 y = Wν = sin(ν arccos(/2)/2) sin(arccos(/2)/2)
3.8 Envelope
3.8.1 Definition
3.8.2 Theorem
3.8.3 fx(g,h,ν)g′ + fy(g,h,ν)h′ + fν(g,h,ν) = 0
3.8.4 fx(g,h,ν)g′ + fy(g,h,ν)h′ = 0
3.8.5 Remark
3.9 The envelope of curve group
3.10 The envelope of curve group
3.10.1 y2arccos2 = arccos2
3.11 The envelope of curve group
3.11.1 y2 sin2(arccos(/2)/2) = 1 or arccos2(/2) = 0