Chapter 46-Equations for Ramanujan Class Invariants and the Ramanujan Ladder

Attached are useful formula for those interested in calculating Ramanujan class invariants and examining the relationships to the properties of the associated octahedron, the Ramanujan octaves and the Ramanujan ladder. Many of the formula are discussed in Chapters 28 to 32 in the Perrin Chalkboard.

Chapter 45- Elliptic Functions, Bell Polynomial and Applications

The previous three chapters discuss the partial Bell Polynomial and its application to integer sequences and various polynomials.  A new formula using the Bell Polynomial to calculate Jacobi elliptic functions is described in this paper.  The infinite polynomial expansion calculates the inverse of the elliptic integrals of the first and second kind.  A single formula for the elliptic sine function, sn, is sufficient to calculate all 12 elliptic functions associated with the elliptic integral of the first kind.  A corresponding formula for the elliptic integral of the second kind is also described.

These functions are used in various applications such as the geometry of ellipses and in astrophysics such as calculation of relativistic elliptical orbits of planets and comets.  Examples of calculated elliptical and hyperbolic orbits such as the precession of the perihelion of the planet Mercury are shown using the Bell Polynomial (sn) function. An association of the Bell Polynomial function to the q octic and the Ramanujan ladder for Weber’s class invariants is discussed

Bell and Elliptic functions

Chapter 30- Ramanujan Octave, Semitones, Chords and Harmonics

Discussion of further investigation of the connection between the octic q fraction, class invariants, tonality and the octahedron.  The Ramanujan ladder is described for any class invariant of discriminant -m. A connection to the volume and edge of the octahedron is proved. This ladder reminds me of Kepler’s third law describing planetary motion found in his Harmonices Mundi (1619) in which the square of the periods (time for one orbit around the sun) of any two planets are proportional to the cubes of their average distance from the sun.

The Ramanujan Octave, Semitones, Chords and Harmonics