Reduce z=x+iy to a congruent value within the
fundamental period parallelogram (FPP). Function
(real, possibly noninteger) m and n such that
Primary complex argument
Vector of length two with first element the first period and second element the second period. Note that p is the period, so p_1=2ω_1, where omega1 is the half period
fpp() is fully vectorized.
mn() to determine the “coordinates” of a
floor(mn(z,p)) %*% p to give the complex value of
the (unique) point in the same period parallelogram as
is congruent to the origin.
Robin K. S. Hankin
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p <- c(1.01+1.123i, 1.1+1.43i) mn(z=1:10,p) %*% p ## should be close to 1:10 #Now specify some periods: p2 <- c(1+1i,1-1i) #Define a sequence of complex numbers that zooms off to infinity: u <- seq(from=0,by=pi+1i*exp(1),len=2007) #and plot the sequence, modulo the periods: plot(fpp(z=u,p=p2)) #and check that the resulting points are within the qpp: polygon(c(-1,0,1,0),c(0,1,0,-1))
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