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If we write the elliptic integral of the first kind in this form:

u = ⌠ ⌡ x 0  dt √   ((1− t2 ) ( 1 − k2 t2 ))   ,

then the elliptic functions sn, cn and dn can be defined as:

x = sn(u) ,

√   1 − x 2   = cn(u) ,

√   1 − k2 x2   = dn(u) .

(This is analogous to defining the circular functions in terms of the integral u = ∫₀ ^x [(dt)/(√{1−t²})] , x=sin(u). )