E
| A page for e, i, π, and φ |
The role that the exponential function (ex) plays is extremely significant to an understanding of the laws of nature and harmonic organisation.
[math]e^x = \sum_{n = 0}^{\infty} {x^n \over n!} = 1 + x + {x^2 \over 2!} + {x^3 \over 3!} + {x^4 \over 4!} + \cdots[/math]
As an infinite sum ex is Σ(xn/n!) = 1 + x / 1 + x.x / 1.2 + x.x.x / 1.2.3 + x.x.x.x / 1.2.3.4 + ...
We can look at the construction of this relation more easily by describing it by an algorithm. We can remove the power and factorial operations by basing the current numerator and denominator values on the previous values:
| +e.pl |
But a better conceptual representation is to restructure it such that the division occurs outside the main loop since its also a higher-level operation. So each iteration is now maintaining the value of a numerator and denominator which represent the entire preceeding series as a single fraction which can be reduced afterwards.
| +alt e.pl |
Here's a snipit in C which uses the taylor series of cosine to make Φ from π
| +phi.c |
Factorial & the Gamma Function
The Gamma function extends the factorial function to complex and non-integer numbers (it is already defined on the naturals, and has simple poles at the negative integers). The Gamma function "fills in" the factorial function for non-integer and complex values of n. If z is a real variable, then for natural number values only, we have
- [math]\Gamma(z+1)= z \times \Gamma(z) = z!\, [/math]
but for non-natural values of z, the above equation does not apply, since the factorial function is not defined.
If the real part of the complex number z is positive, then the following integral converges absolutely:
- [math]\Gamma(z) = \int_0^\infty t^{z-1} e^{-t}\,dt[/math]
An interesting property is the Gamma functions link to π;
- [math] \Gamma(\frac{1}{2}) = \sqrt\pi [/math]
Taylor's theorem expresses a function f(x) as a power series in x, basically because the nth derivative of xn is n!.
- The volume of an n-dimensional hypersphere can be expressed as:
- [math]V_n={\pi^{n/2}R^n\over (n/2)!}.[/math]
Note that the Gamma function is required for odd dimensions and that its value cancels out the apparent fractional power of π in those cases.
- See also
