# Calculates the invariants g2 and g3

### Description

Calculates the invariants g2 and g3 using any of a number of methods

### Usage

1 2 3 4 5 6 7 8 9 10 11 | ```
g.fun(b, ...)
g2.fun(b, use.first=TRUE, ...)
g3.fun(b, use.first=TRUE, ...)
g2.fun.lambert(b, nmax=50, tol=1e-10, strict=TRUE)
g3.fun.lambert(b, nmax=50, tol=1e-10, strict=TRUE)
g2.fun.direct(b, nmax=50, tol=1e-10)
g3.fun.direct(b, nmax=50, tol=1e-10)
g2.fun.fixed(b, nmax=50, tol=1e-10, give=FALSE)
g3.fun.fixed(b, nmax=50, tol=1e-10, give=FALSE)
g2.fun.vectorized(b, nmax=50, tol=1e-10, give=FALSE)
g3.fun.vectorized(b, nmax=50, tol=1e-10, give=FALSE)
``` |

### Arguments

`b` |
Half periods. NB: the arguments
are the |

`nmax` |
Maximum number of terms to sum. See details section for more discussion |

`tol` |
Numerical tolerance for stopping: summation stops when adding an additional term makes less |

`strict` |
Boolean, with default (where taken) |

`give` |
Boolean, with default (where taken) |

`...` |
In functions |

`use.first` |
In function |

### Details

Functions `g2.fun()`

and `g3.fun()`

use theta functions
which converge very quickly. These functions are the best in most
circumstances. The theta functions include a loop that continues to add
terms until the partial sum is unaltered by addition
of the next term. Note that summation continues until *all*
elements of the argument are properly summed, so performance is
limited by the single worst-case element.

The following functions are provided for interest only, although there is a remote possibility that some weird circumstances may exist in which they are faster than the theta function approach.

Functions `g2.fun.divisor()`

and `g3.fun.divisor()`

use
Chandrasekharan's formula on page 83. This is generally slower than
the theta function approach

Functions `g2.fun.lambert()`

and `g3.fun.lambert()`

use a
Lambert series to accelerate Chandrasekharan's formula. In general,
it is a little better than the divisor form.

Functions `g2.fun.fixed()`

and `g2.fun.fixed()`

also use
Lambert series. These functions are vectorized in the sense that
the function body uses only vector operations. These functions do
not take a vector argument. They are called “fixed” because
the number of terms used is fixed in advance (unlike `g2.fun()`

and `g3.fun()`

).

Functions `g2.fun.vectorized()`

and `g3.fun.vectorized()`

also use Lambert series. They are fully vectorized in that they take
a vector of periods or period ratios, unlike the previous two
functions. However, this can lead to loss of precision in some
cases (specifically when the periods give rise to widely varying
values of g2 and g3).

Functions `g2.fun.direct()`

and `g3.fun.direct()`

use a
direct summation. These functions are absurdly slow. In general,
the Lambert series functions converge much faster; and the
“default” functions `g2.fun()`

and `g3.fun()`

,
which use theta functions, converge faster still.

### Author(s)

Robin K. S. Hankin

### References

Mathematica website

### Examples

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 | ```
g.fun(half.periods(g=c(8,4+1i))) ## should be c(8,4+1i)
## Example 4, p664, LHS:
omega <- c(10,11i)
(g2 <- g2.fun(omega))
(g3 <- g3.fun(omega))
e1e2e3(Re(c(g2,g3)))
## Example 4, p664, RHS:
omega2 <- 10
omega2dash <- 11i
omega1 <- (omega2-omega2dash)/2 ## From figure 18.1, p630
(g2 <- g2.fun(c(omega1,omega2)))
(g3 <- g3.fun(c(omega1,omega2)))
e1e2e3(Re(c(g2,g3)))
``` |

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