Neville's form for the theta functions — theta.neville • elliptic

Neville's notation for theta functions as per section 16.36 of Abramowitz and Stegun.

theta.s(u, m, method = "16.36.6", ...)
theta.c(u, m, method = "16.36.6", ...)
theta.d(u, m, method = "16.36.7", ...)
theta.n(u, m, method = "16.36.7", ...)

Arguments

u: Primary complex argument
m: Real parameter
method: Character string corresponding to A and S's equation numbering scheme
...: Extra arguments passed to the method function, such as maxiter

Details

I reproduce the relevant sections of AMS-55 here, for convenience:

	16.36.6a
\(\displaystyle\vartheta_s(u) = \frac{2K\vartheta_1(v)}{\vartheta'_{1_{\vphantom{j_j}}}(0)}\)	16.36.6b
\(\displaystyle\vartheta_c(u) = \frac{\vartheta_2(v) }{\vartheta _{2_{\vphantom{j_j}}}(0)}\)	16.36.7a
\(\displaystyle\vartheta_d(u) = \frac{\vartheta_3(v) }{\vartheta _{3_{\vphantom{j_j}}}(0)}\)	16.36.7b
\(\displaystyle\vartheta_n(u) = \frac{\vartheta_4(v) }{\vartheta _{4_{\vphantom{j_j}}}(0)}\)	16.37.1
\(\displaystyle\vartheta_s(u)=\left(\frac{16q}{mm_1}\right)^{1/6}\sin v\prod_{n=1}^\infty\left(1-2q^{2n}\cos 2v+q^{4n}\right)\)	16.37.2
\(\displaystyle\vartheta_c(u)=\left(\frac{16qm_1^{1/2}}{m}\right)^{1/6}_{\vphantom{j_j}}\cos v\prod_{n=1}^\infty\left(1+2q^{2n}\cos 2v+q^{4n}\right)\)	16.37.3
\(\displaystyle\vartheta_d(u)=\left(\frac{mm_1}{16q}\right)^{1/12} \prod_{n=1}^\infty\left(1+2q^{2n-1}\cos 2v+q^{4n-2}\right)\)	16.37.4

(in the above we have \(v=\pi u/(2K)\) and \(q=q(m)\)).

References

M. Abramowitz and I. A. Stegun 1965. Handbook of mathematical functions. New York: Dover

Author

Robin K. S. Hankin

Examples

#Figure 16.4.
m <- 0.5
K <- K.fun(m)
Kdash <- K.fun(1-m)
x <- seq(from=0,to=4*K,len=100)
plot  (x/K,theta.s(x,m=m),type="l",lty=1,main="Figure 16.4, p578")
points(x/K,theta.n(x,m=m),type="l",lty=2)
points(x/K,theta.c(x,m=m),type="l",lty=3)
points(x/K,theta.d(x,m=m),type="l",lty=4)
abline(0,0)




#plot a graph of something that should be zero:
 x <- seq(from=-4,to=4,len=55)
 plot(x,(e16.37.1(x,0.5)-theta.s(x,0.5)),pch="+",main="error: note vertical scale")


#now table 16.1 on page 582 et seq:
 alpha <- 85
 m <- sin(alpha*pi/180)^2
## K <- ellint_Kcomp(sqrt(m))
 K <- K.fun(m)
 u <- K/90*5*(0:18)
 u.deg <- round(u/K*90)
 cbind(u.deg,"85"=theta.s(u,m))      # p582, last col. 
#>       u.deg        85
#>  [1,]     0 0.0000000
#>  [2,]     5 0.2132177
#>  [3,]    10 0.4284434
#>  [4,]    15 0.6474349
#>  [5,]    20 0.8714648
#>  [6,]    25 1.1011162
#>  [7,]    30 1.3361236
#>  [8,]    35 1.5752683
#>  [9,]    40 1.8163399
#> [10,]    45 2.0561678
#> [11,]    50 2.2907234
#> [12,]    55 2.5152906
#> [13,]    60 2.7246942
#> [14,]    65 2.9135742
#> [15,]    70 3.0766867
#> [16,]    75 3.2092122
#> [17,]    80 3.3070473
#> [18,]    85 3.3670599
#> [19,]    90 3.3872870
 cbind(u.deg,"85"=theta.n(u,m))      # p583, last col. 
#>       u.deg       85
#>  [1,]     0 1.000000
#>  [2,]     5 1.016639
#>  [3,]    10 1.066184
#>  [4,]    15 1.147516
#>  [5,]    20 1.258756
#>  [6,]    25 1.397253
#>  [7,]    30 1.559573
#>  [8,]    35 1.741516
#>  [9,]    40 1.938152
#> [10,]    45 2.143900
#> [11,]    50 2.352647
#> [12,]    55 2.557921
#> [13,]    60 2.753098
#> [14,]    65 2.931653
#> [15,]    70 3.087425
#> [16,]    75 3.214899
#> [17,]    80 3.309465
#> [18,]    85 3.367648
#> [19,]    90 3.387287