Proofs of fast formulas for computing constant pi

• Nedeljko
In summary, the conversation discusses the request for information on mathematical backgrounds of fast algorithms for computing the number pi, specifically the Gauss Legendre algorithm, Borwein algorithm, Ramanujan formulas, and Chudnovsky formula. The speaker also mentions being interested in the relation between arithmetic-geometric mean and complete elliptic integrals of the first kind, and requests for online sources or proofs on this topic.
Nedeljko
I am interesting for mathematical background od fast algorithms for computing number $$\pi$$ with complete proofs only. More specific, I am interesting for Gauss Legendre algorithm, Borwein algorithm, Ramanujan formulas and Chudnovsky formula.

Have you considered buying a text-book?

I am skillful in general mathematics. About Gauss Legendre formula, how to prove relation between arithmetic-geometric mean and complete elliptic integral of the first kind? I proved that it is equivalent to formula $$K(\sin^2(2x))\cos^2x=K(\tan^4x)$$, where $$K$$ is the complete elliptic integral of the first kind.

Nedeljko said:
I've searched a bit online, but haven't, as yet, dug up any. Hopefully, somebody else might lead you there.

Does somebody has ideas for proofs or links?

1. How is the constant pi calculated?

The constant pi is calculated using mathematical formulas that are designed to approximate its value. These formulas often involve infinite series or limits, and are used to find more and more accurate approximations of pi.

2. What are some common fast formulas for computing pi?

Some common fast formulas for computing pi include the Gregory-Leibniz series, the Nilakantha series, and the Chudnovsky algorithm. These formulas are designed to converge quickly, providing more accurate approximations of pi in fewer steps.

3. How do these fast formulas compare to other methods of computing pi?

Faster formulas for computing pi are typically more efficient than other methods, such as using geometric shapes or Monte Carlo simulations. They require less computational power and can yield more accurate results in a shorter amount of time.

4. Can these fast formulas be used to calculate pi to an exact value?

No, fast formulas for computing pi can only provide approximations of the constant. They may be able to calculate pi to a very high degree of accuracy, but they will never be able to provide an exact value.

5. How are these fast formulas used in real-world applications?

Fast formulas for computing pi are used in various fields such as engineering, physics, and computer science. They are essential for calculations involving circular shapes or processes, and are also used in algorithms for data compression and encryption.

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