theta.neville | R Documentation |
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", ...)
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
|
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 | \displaystyle\vartheta_n(u)=\left(\frac{m}{16qm_1^2}\right)^{1/12}
\prod_{n=1}^\infty\left(1-2q^{2n-1}\cos 2v+q^{4n-2}\right)
|
(in the above we have v=\pi u/(2K)
and q=q(m)
).
Robin K. S. Hankin
M. Abramowitz and I. A. Stegun 1965. Handbook of mathematical functions. New York: Dover
#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.
cbind(u.deg,"85"=theta.n(u,m)) # p583, last col.
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