Coefficients of Laurent expansion of Weierstrass P function

Calculates the coefficients of the Laurent expansion of the Weierstrass \(\wp\) function in terms of the invariants

ck(g, n=20)

Arguments

g

The invariants: a vector of length two with g=c(g2,g3)

n

length of series

Details

Calculates the series \(c_k\) as per equation 18.5.3, p635.

Author

Robin K. S. Hankin

See also

P.laurent

Examples

 #Verify 18.5.16, p636:
 x <- ck(g=c(0.1+1.1i,4-0.63i))
14*x[2]*x[3]*(389*x[2]^3+369*x[3]^2)/3187041-x[11] #should be zero
#> [1] 2.646978e-23+5.293956e-23i


# Now try a random example by comparing the default (theta function) method
# for P(z) with the Laurent expansion:

z <- 0.5-0.3i
g <- c(1.1-0.2i, 1+0.4i)
series <- ck(15,g=g)
1/z^2+sum(series*(z^2)^(0:14)) - P(z,g=g) #should be zero
#> [1] 2.664535e-15-4.440892e-16i