In mathematics, the Neville theta functions, named after Eric Harold Neville,[1] are defined as follows:[2] [3] [4]
| 1/4 | ||||
| \theta | ||||
|
| \theta | ||||
|
\thetan(z,m)=
| \sqrt{2\pi | |
\thetas(z,m)=
| \sqrt{2\pi | |
| q(m) |
1/4
where: K(m) is the complete elliptic integral of the first kind,
K'(m)=K(1-m)
q(m)=e-\pi
Note that the functions θp(z,m) are sometimes defined in terms of the nome q(m) and written θp(z,q) (e.g. NIST). The functions may also be written in terms of the τ parameter θp(z|τ) where
q=ei\pi\tau
The Neville theta functions may be expressed in terms of the Jacobi theta functions[5]
\thetas(z|\tau)=\theta
| 2(0|\tau)\theta | |
| 1(z'|\tau)/\theta' |
1(0|\tau)
\thetac(z|\tau)=\theta2(z'|\tau)/\theta2(0|\tau)
\thetan(z|\tau)=\theta4(z'|\tau)/\theta4(0|\tau)
\thetad(z|\tau)=\theta3(z'|\tau)/\theta3(0|\tau)
where
| 2(0|\tau) | |
| z'=z/\theta | |
| 3 |
The Neville theta functions are related to the Jacobi elliptic functions. If pq(u,m) is a Jacobi elliptic function (p and q are one of s,c,n,d), then
| \operatorname{pq}(u,m)= | \thetap(u,m) |
| \thetaq(u,m) |
.
\thetac(2.5,0.3) ≈ -0.65900466676738154967
\thetad(2.5,0.3) ≈ 0.95182196661267561994
\thetan(2.5,0.3) ≈ 1.0526693354651613637
\thetas(2.5,0.3) ≈ 0.82086879524530400536
\thetac(z,m)=\thetac(-z,m)
\thetad(z,m)=\thetad(-z,m)
\thetan(z,m)=\thetan(-z,m)
\thetas(z,m)=-\thetas(-z,m)