Theta Function - Explicit Values

Explicit Values

See

$\varphi(e^{-\pi x}) = \vartheta(0; {\mathrm{i}}x) = \theta_3(0;e^{-\pi x}) = \sum_{n=-\infty}^\infty e^{-x \pi n^2}$

$\varphi\left(e^{-\pi} \right) = \frac{\sqrt{\pi}}{\Gamma(\frac{3}{4})}$

$\varphi\left(e^{-2\pi} \right) = \frac{\sqrt{6\pi+4\sqrt2\pi}}{2\Gamma(\frac{3}{4})}$

$\varphi\left(e^{-3\pi}\right) = \frac{\sqrt{27\pi+18\sqrt3\pi}}{3\Gamma(\frac{3}{4})}$

$\varphi\left(e^{-4\pi}\right) =\frac{\sqrt{8\pi}+2\sqrt{\pi}}{4\Gamma(\frac{3}{4})}$

$\varphi\left(e^{-5\pi} \right) =\frac{\sqrt{225\pi+ 100\sqrt5 \pi}}{5\Gamma(\frac{3}{4})}$

$\varphi\left(e^{-6\pi}\right) = \frac{\sqrt{3\sqrt{2}+3\sqrt{3}+2\sqrt{3}-\sqrt{27}+\sqrt{1728}-4}\cdot \sqrt{243{\pi}^2}}{6\sqrt{1+\sqrt6-\sqrt2-\sqrt3}{\Gamma(\frac{3}{4})}}$

Read more about this topic: Theta Function

Famous quotes containing the words explicit and/or values:

“Like dreaming, reading performs the prodigious task of carrying us off to other worlds. But reading is not dreaming because books, unlike dreams, are subject to our will: they envelop us in alternative realities only because we give them explicit permission to do so. Books are the dreams we would most like to have, and, like dreams, they have the power to change consciousness, turning sadness to laughter and anxious introspection to the relaxed contemplation of some other time and place.”
—Victor Null, South African educator, psychologist. Lost in a Book: The Psychology of Reading for Pleasure, introduction, Yale University Press (1988)

“We cannot spare our children the influence of harmful values by turning off the television any more than we can keep them home forever or revamp the world before they get there. Merely keeping them in the dark is no protection and, in fact, can make them vulnerable and immature.”
—Polly Berrien Berends (20th century)