
1. Laplacian over Hyperbolic Prolate Spheroidal Coordinates
For the sphere, typically, the procedure of the conversion of Laplacian from Cartesian(XYZ) to Polar Coordinates systems is via Cylindrical Coordinates. For the spheroid, the same procedure is useable too. And the hyperbolic function is useable too.
Here, for a spheroid, the conversion of Laplacian from Cartesian to hyperbolic Prolate Spheroidal Coordinates systems was carried out via hyperbolic Elliptic Cylindrical Coordinates.
At first, the Laplacian of hyperbolic Elliptic Cylindrical Coordinates was calculated.
And next(finally), the Laplacian of hyperbolic Prolate Spheroidal Coordinates was calculated.
1.1 Hyperbolic Elliptic Cylindrical Coordinates (μ_{1}, ν_{1}, y_{1})
The coordinates hyperbolic function cosh_{ }μ_{1} and cosine function cos_{ }ν_{1} are given by the equations below.
The range of the variable μ_{1} is from 0 to ∞ and the range of the variable ν_{1} is from 0 to 2π. Right figure shows XZ plane over Cartesian coordinates, in which electron 1 is point P_{1} (x_{1}, y_{1}, z_{1}), nuclear A is point A (0, 0, a), and nuclear B is point B (0, 0, a) over Cartesian coordinates. r_{AB} or 2a is the distance of nuclear A from nuclear B, and is constant in accordance with Born Oppenheimer approximation in the electronic orbital of Schrödinger equation. Therefore nuclear A and B are fixed on point A and B, respectively. In the other side, electron 1 is variable on XZ plane. And r_{A1} and r_{B1} is the distance of nuclear A and B, respectively, from electron 1.
From the Pitagorass theorem of the rightangled triangle A z’_{1}P_{1} and B z’_{1}P_{1}, z_{1} and x_{1} appear below.


Below equations show the correlations between cosh μ_{1} and cos_{ }ν_{1}, over hyperbolic Elliptic Cylindrical coordinates, and r_{A1} and r_{B1}, over Cartesian coordinates.
The formulas of the hyperbolic function and the trigonometric function
Hyperbolic function
The hyperbolic functions are definited as below, but these formulas are not used here.
Trigonometric function
Ellipsoid on XZ plane is moved along Y axias. The hyperbolic Elliptic Cylindrical Coordinates is below.
The ranges of the variables are below.

1.2 Laplacian over hyperbolic Elliptic Cylindrical Coordinates
1.3 Hyperbolic Prolate Spheroidal Coordinates
The coordinates hyperbolic function cosh μ_{1} and cosine function cos_{ }ν_{1} are also given by the equations below.
The range of the variable μ_{1} is from 0 to ∞ and the range of the variable ν_{1} is from 0 to π. Right figure shows XZ plane (_{1}=0) over Cartesian coordinates, the coordinates is same with Hyperbolic Elliptic Cylindrical Coordinates except x_{1}→ρ_{1}.
From the Pitagorass theorem of the rightangled triangle A z’_{1}P_{1} and B z’_{1}P_{1}, z_{1} and ρ_{1} appear also below.


In the next step, the plane involving triangle A B P_{1} is rotated on Z axis, and the rotation angle is _{1}. The range of the variable _{1} is from 0 to 2π.
Right figure shows XY plane over Cartesian coordinates. We draw a perpendicular line from P_{1} to X axis, and the point of intersection is a point x’_{1}. Further we draw a perpendicular line from P_{1} to Y axis , and the point of intersection is a point y’_{1}. The point x’_{1} and y’_{1} are (x_{1},0,0) and (0,y_{1},0), respectively.
The Hyperbolic Prolate Spheroidal Coordinates is below.
The ranges of the variables are below.



1.4 Laplacian over hyperbolic Prolate Spheroidal Coordinates

