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Second order linear homogeneous differential equations which are satisfied by polynomials English edition

Second order linear homogeneous differential equations which are satisfied by polynomials English edition

Yukitaka Miyagawa
0/5 ( ratings)
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

Yukitaka Miyagawa
0/5 ( ratings)
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

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