The Case With 1 A Basic Half-period
If, much of the above theory becomes simpler; it is then conventional to write for . For a fixed τ in the upper half-plane, so that the imaginary part of τ is positive, we define the Weierstrass ℘ function by
The sum extends over the lattice {n+mτ : n and m in Z} with the origin omitted. Here we regard τ as fixed and ℘ as a function of z; fixing z and letting τ vary leads into the area of elliptic modular functions.
Read more about this topic: Weierstrass's Elliptic Functions
Famous quotes containing the words case and/or basic:
“Never in any case say I have lost such a thing, but I have returned it. Is your child dead? It is a return. Is your wife dead? It is a return. Are you deprived of your estate? Is not this also a return?”
—Epictetus (c. 55–135)
“What, then, is the basic difference between today’s computer and an intelligent being? It is that the computer can be made to see but not to perceive. What matters here is not that the computer is without consciousness but that thus far it is incapable of the spontaneous grasp of pattern—a capacity essential to perception and intelligence.”
—Rudolf Arnheim (b. 1904)