Definition in Terms of Theta Functions
Equivalently, Jacobi elliptic functions can be defined in terms of his theta functions. If we abbreviate as, and respectively as (the theta constants) then the elliptic modulus k is . If we set, we have
Since the Jacobi functions are defined in terms of the elliptic modulus k(τ), we need to invert this and find τ in terms of k. We start from, the complementary modulus. As a function of τ it is
Let us first define
Then define the nome q as and expand as a power series in the nome q, we obtain
Reversion of series now gives
Since we may reduce to the case where the imaginary part of τ is greater than or equal to 1/2 sqrt(3), we can assume the absolute value of q is less than or equal to exp(-1/2 sqrt(3) π) ~ 0.0658; for values this small the above series converges very rapidly and easily allows us to find the appropriate value for q.
Read more about this topic: Jacobi Elliptic Functions
Famous quotes containing the words definition in, definition, terms and/or functions:
“Was man made stupid to see his own stupidity?
Is God by definition indifferent, beyond us all?
Is the eternal truth man’s fighting soul
Wherein the Beast ravens in its own avidity?”
—Richard Eberhart (b. 1904)
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)
“My father and I were always on the most distant terms when I was a boy—a sort of armed neutrality, so to speak. At irregular intervals this neutrality was broken, and suffering ensued; but I will be candid enough to say that the breaking and the suffering were always divided up with strict impartiality between us—which is to say, my father did the breaking, and I did the suffering.”
—Mark Twain [Samuel Langhorne Clemens] (1835–1910)
“When Western people train the mind, the focus is generally on the left hemisphere of the cortex, which is the portion of the brain that is concerned with words and numbers. We enhance the logical, bounded, linear functions of the mind. In the East, exercises of this sort are for the purpose of getting in tune with the unconscious—to get rid of boundaries, not to create them.”
—Edward T. Hall (b. 1914)