Elliptic Integral - Incomplete Elliptic Integral of The Third Kind

The incomplete elliptic integral of the third kind Π is

\Pi(n ; \varphi \setminus \alpha) = \int_0^\varphi \frac{1}{1-n\sin^2 \theta} \frac {d\theta}{\sqrt{1-(\sin\theta\sin \alpha)^2}}, or
\Pi(n ; \varphi \,|\,m) = \int_{0}^{\sin \varphi} \frac{1}{1-nt^2} \frac{dt}{\sqrt{(1-m t^2)(1-t^2) }}.

The number n is called the characteristic and can take on any value, independently of the other arguments. Note though that the value is infinite, for any m.

A relation with the Jacobian elliptic functions is

The meridian arc length from the equator to latitude is also related to a special case of Π:

Read more about this topic:  Elliptic Integral

Famous quotes containing the words incomplete, integral and/or kind:

    “Someone once asked me why women don’t gamble as much as men do, and I gave the common-sensical reply that we don’t have as much money. That was a true but incomplete answer. In fact, women’s total instinct for gambling is satisfied by marriage.”
    —Gloria Steinem (b. 1934)

    “Self-centeredness is a natural outgrowth of one of the toddler’s major concerns: What is me and what is mine...? This is why most toddlers are incapable of sharing ... to a toddler, what’s his is what he can get his hands on.... When something is taken away from him, he feels as though a piece of him—an integral piece—is being torn from him.”
    —Lawrence Balter (20th century)

    “Bonnie Lee: Oh, it’s the most wonderful thing I’ve ever seen.
    Geoff Carter [sarcastically]: Yes, it reminded you of a great big, beautiful bird, didn’t it?
    Bonnie: No, it didn’t at all. That’s why it’s so wonderful. It’s really a flying human being.
    Geoff: Well, you’re right about one thing. A bird’d have too much sense to fly in that kind of muck.”
    —Jules Furthman (1888–1960)