User:Guardian of Light

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User at Wikipedia.

Major Contributions[edit]

Some pages I've invested a lot in:

  1. Differentiable manifold
  2. Proofs of Fermat's little theorem
  3. John Derbyshire
  4. Ayla (Chrono Trigger)
  5. Frog (Chrono Trigger)
  6. Sonic Heroes

Minor Contributions[edit]

Mathematical[edit]

  1. Trigonometric function
  2. Hyperbolic function
  3. Taylor series
  4. Taylor's theorem
  5. Table of integrals
  6. Derivative
  7. Partial fraction
  8. Metamathematics
  9. Coversine
  10. Critical line theorem
  11. Fermat's little theorem
  12. Limit (mathematics)
  13. Domain (mathematics)
  14. Laplace transform
  15. 173 (number)
  16. Amortization (business)

Scientific[edit]

  1. Carl Friedrich Gauss
  2. Chaos theory
  3. Wave equation
  4. P-branes
  5. Great White Shark


Miscellaneous[edit]

  1. Chrono Trigger
  2. Melchior (Chrono Trigger)
  3. English Language
  4. Crono
  5. Serge (Chrono Cross)
  6. Kid (Chrono Cross)
  7. Lynx (Chrono Cross)
  8. Luccia
  9. Harle

Creations[edit]

The pages I myself have made (from ye olde scratch) are--in chronological order:

  1. Haversine
  2. This Page
  3. Heliotrope
  4. Magnometer
  5. Belthasar (Chrono Trigger)
  6. 12000 B.C. (Chrono Trigger)

Problems[edit]

To show the probability that two integers chosen at random are relatively prime is {6\over \pi^2}.

Proof: It is sufficient to show \sum_{n=1}^{\infty}{1\over n^2} = {\pi^2\over 6}. When we have a polynomial with constant term one, we may rewrite it in factored form as follows: If \alpha_1,\alpha_2,...,\alpha_r\, are the roots of a polynomial p(z), then we may write p(z)=\left(1-{z\over\alpha_1}\right)...\left(1-{z\over \alpha_r}\right).

Now examine the power series for the function sin(z)/z. {\sin z\over z}=1-{z^2\over 3!}+{z^4\over 5!}+...+{(-1)^nz^{2n}\over (2n+1)!}

Well we also know we can rewrite sin(z)/z in terms of its roots to be:

\left(1-{z\over \pi}\right)\left(1+{z\over \pi}\right)\left(1-{z\over 2\pi}\right)\left(1+{z\over 2\pi}\right)\left(1-{z\over 3\pi}\right)\left(1+{z\over 3\pi}\right)...\left(1-{z\over k\pi}\right)\left(1+{z\over k\pi}\right)...

If we examine the quadratic term in each we find that:

{1\over 3!} = {1\over \pi^2}\sum_{n = 1}^{\infty}{1\over n^2}\rightarrow {\pi^2\over 6}=\sum_{n = 1}^{\infty}{1\over n^2}\text{Q.E.D.}