theta | R Documentation |
Computes Jacobi's four theta functions for complex z
in terms
of the parameter m
or q
.
theta1 (z, ignore=NULL, m=NULL, q=NULL, give.n=FALSE, maxiter=30, miniter=3)
theta2 (z, ignore=NULL, m=NULL, q=NULL, give.n=FALSE, maxiter=30, miniter=3)
theta3 (z, ignore=NULL, m=NULL, q=NULL, give.n=FALSE, maxiter=30, miniter=3)
theta4 (z, ignore=NULL, m=NULL, q=NULL, give.n=FALSE, maxiter=30, miniter=3)
theta.00(z, ignore=NULL, m=NULL, q=NULL, give.n=FALSE, maxiter=30, miniter=3)
theta.01(z, ignore=NULL, m=NULL, q=NULL, give.n=FALSE, maxiter=30, miniter=3)
theta.10(z, ignore=NULL, m=NULL, q=NULL, give.n=FALSE, maxiter=30, miniter=3)
theta.11(z, ignore=NULL, m=NULL, q=NULL, give.n=FALSE, maxiter=30, miniter=3)
Theta (u, m, ...)
Theta1(u, m, ...)
H (u, m, ...)
H1(u, m, ...)
z,u |
Complex argument of function |
ignore |
Dummy variable whose intention is to force the user to
name the second argument either |
m |
Does not seem to have a name. The variable is introduced in section 16.1, p569 |
q |
The nome |
give.n |
Boolean with default |
maxiter |
Maximum number of iterations used. Note that the series generally converge very quickly |
miniter |
Minimum number of iterations to guard against premature exit if an addend is zero exactly |
... |
In functions that take it, extra arguments passed to
|
Functions theta.00()
et seq are just wrappers for
theta1()
et seq, following Whittaker and Watson's terminology
on p487; the notation does not appear in Abramowitz and Stegun.
theta.11() = theta1()
theta.10() = theta2()
theta.00() = theta3()
theta.01() = theta4()
Returns a complex-valued object with the same attributes as either
z
, or (m
or q
), whichever wasn't recycled.
Robin K. S. Hankin
M. Abramowitz and I. A. Stegun 1965. Handbook of mathematical functions. New York: Dover
theta.neville
m <- 0.5
derivative <- function(small){(theta1(small,m=m)-theta1(0,m=m))/small}
right.hand.side1 <- theta2(0,m=m)*theta3(0,m=m)*theta4(0,m=m)
right.hand.side2 <- theta1.dash.zero(m)
derivative(1e-5) - right.hand.side1 # should be zero
derivative(1e-5) - right.hand.side2 # should be zero
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