Difference between revisions of "E"

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[[Category:nad]][[Category:philosophy]]
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{{lowercase}}__NOTOC__
 
<table width=100%><tr><td align=center><font size=5>A page for e, i, &pi;, and &phi;</font></table>
 
<table width=100%><tr><td align=center><font size=5>A page for e, i, &pi;, and &phi;</font></table>
  
 +
[[Image:EApprox.png|thumb|300px|Approximation of e<sup>x</sup> with first four [[W:Taylor series|Maclaurin series]] polynomials]]
 
The role that the exponential function (''e<sup>x</sup>'') plays is extremely significant to an understanding of the laws of nature and harmonic organisation.
 
The role that the exponential function (''e<sup>x</sup>'') plays is extremely significant to an understanding of the laws of nature and harmonic organisation.
 +
Mathematically (''e<sup>x</sup>'') can be expressed several ways. A common form is the [[W:Taylor series|Maclaurin series]] expansion of;
  
 
<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>
 
<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>
  
 +
Note that;
  
[[Image:Riemann II.jpg]]
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<math>e^1 = \sum_{n=0}^{\infty}{1^n \over n!} = 1 + 1 + {1^2 \over 2!} + {1^3 \over 3!} + {1^4 \over 4!} + \cdots</math>
 +
 
 +
and
 +
 
 +
<math>\frac{1}{e} = \sum_{n=0}^{\infty} (-1)^n\frac{1}{n!} = 1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\cdots</math>
 +
 
 +
where
 +
 
 +
<math> e^2 = e^{(1+1)} = e^1 \times e^1 </math>.
 +
 
 +
or
 +
 
 +
<math> \sum_{n=0}^{\infty}{2^n \over n!} = \sum_{n=0}^{\infty}{1^n \over n!} \times \sum_{m=0}^{\infty}{1^m \over m!}</math>.
 +
 
 +
Expansion of the right hand side terms is an [[W:Outer product|outer product]], taking the summation of these terms equals e<sup>2</sup>.
 +
 
 +
[[Image:Riemann II.jpg|thumb|200px|This formula needs to be checked and converted to LaTeX]]
 
*[http://en.wikipedia.org/wiki/Srinavasa_Ramanujan Srinavasa Ramanujan]
 
*[http://en.wikipedia.org/wiki/Srinavasa_Ramanujan Srinavasa Ramanujan]
  
As an infinite sum  [[Wikipedia:Exponential function|e<sup>x</sup>]] is ''&Sigma;(x<sup>n</sup>/n!)'' &nbsp; = &nbsp; 1 &nbsp;
+
As an infinite sum  [[Wikipedia:Exponential function|e<sup>x</sup>]] is;
+ &nbsp; ''x'' / 1 &nbsp;
+
 
+ &nbsp; ''x.x'' / 1.2 &nbsp;
+
<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>
+ &nbsp; ''x.x.x'' / 1.2.3 &nbsp;
 
+ &nbsp; ''x.x.x.x'' / 1.2.3.4 + ... &nbsp;
 
  
 
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:
 
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:
<table class=document-code><tr><td>[[+e.pl]]</table>
+
{{: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.
 
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.
<table class=document-code><tr><td>[[+alt e.pl]]</table>
+
{{:alt e.pl}}
  
 
Here's a snipit in C which uses the taylor series of cosine to make &Phi; from &pi;
 
Here's a snipit in C which uses the taylor series of cosine to make &Phi; from &pi;
<table class=document-code><tr><td>[[+phi.c]]</table>
+
{{:phi.c}}
  
=== The unit circle ===
+
== The unit circle ==
 
The ''n''<sup>th</sup> [[w:roots of unity|roots of unity]], or [[w:Abraham de Moivre|de Moivre]] numbers, are all the complex numbers which yield 1 when raised to a given power ''n''. It can be shown that they are located on the [[w:unit circle|unit circle]] of the complex plane and that in that plane they form the vertices of a ''n''-sided regular polygon with one vertex on 1.
 
The ''n''<sup>th</sup> [[w:roots of unity|roots of unity]], or [[w:Abraham de Moivre|de Moivre]] numbers, are all the complex numbers which yield 1 when raised to a given power ''n''. It can be shown that they are located on the [[w:unit circle|unit circle]] of the complex plane and that in that plane they form the vertices of a ''n''-sided regular polygon with one vertex on 1.
  
=== Factorial & the Gamma Function ===
+
== Factorial & the Gamma Function ==
 
The [[w:Gamma function|Gamma function]] extends the [[w:factorial|factorial]] function to [[w:complex number|complex]] and non-integer numbers (it is already defined on the naturals, and has [[w:simple pole|simple pole]]s 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
 
The [[w:Gamma function|Gamma function]] extends the [[w:factorial|factorial]] function to [[w:complex number|complex]] and non-integer numbers (it is already defined on the naturals, and has [[w:simple pole|simple pole]]s 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>
+
<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.
 
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 [[w:Absolute convergence|converges absolutely]]:
 
If the real part of the complex number ''z'' is positive, then the following integral [[w:Absolute convergence|converges absolutely]]:
:<math>\Gamma(z) = \int_0^\infty  t^{z-1} e^{-t}\,dt</math>
+
 
 +
<math>\Gamma(z) = \int_0^\infty  t^{z-1} e^{-t}\,dt</math>
 +
 
 
An interesting property is the [[w:Gamma function|Gamma function]]s link to &pi;;
 
An interesting property is the [[w:Gamma function|Gamma function]]s link to &pi;;
:<math> \Gamma(\frac{1}{2}) = \sqrt\pi </math>
+
 
----
+
<math> \Gamma(\frac{1}{2}) = \sqrt\pi </math>
 +
 
 +
== Taylor series expansion ==
 
[[w:Taylor's theorem|Taylor's theorem]] expresses a function ''f''(''x'') as a power series in ''x'', basically because the ''n''<sup>th</sup> derivative of ''x''<sup>''n''</sup> is ''n''!.
 
[[w:Taylor's theorem|Taylor's theorem]] expresses a function ''f''(''x'') as a power series in ''x'', basically because the ''n''<sup>th</sup> derivative of ''x''<sup>''n''</sup> is ''n''!.
 
:Why is it that when we reduce the function down to degree-1 through differentiation it ends up as n!? --[[User:Nad|Nad]] 17:16, 26 Oct 2006 (NZDT)
 
:Why is it that when we reduce the function down to degree-1 through differentiation it ends up as n!? --[[User:Nad|Nad]] 17:16, 26 Oct 2006 (NZDT)
----
+
::Because each derivative multiplies by the power and reduces the power by one. --[[User:Nad|Nad]] 12:12, 12 July 2007 (NZST)
* The volume of an ''n''-dimensional hypersphere can be expressed as:
+
:::So maybe the ''n!'' denominator in talor-series of ''e<sup>x</sup>'' should be thought of as the ''n<sup>th</sup>'' derivative rather than as combinatorial. --[[User:Nad|Nad]] 12:16, 12 July 2007 (NZST)
 +
Actually, the [[W:Taylor series|Taylor series]] article explains it - the theorem in general for a function ''f(a)'' is,
 +
 
 +
<math>\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^{n}\,,</math>
  
::<math>V_n={\pi^{n/2}R^n\over (n/2)!}.</math>
+
where ''f''&nbsp;<sup>(''n'')</sup>(''a'') denotes the ''n''th derivative of ''f'' at the point ''a''; the zeroth derivative of ''f'' is defined to be ''f'' itself and <math>(x-a)^0</math> is defined to be 1. But all the derivatives are the still ''e<sup>x</sup>'' for the exponential function, and we're using ''a=0'', so the top term here is always equal to 1, leaving only ''x<sup>n</sup>'' on the top. --[[User:Nad|Nad]] 18:22, 12 July 2007 (NZST)
 +
 
 +
== Spheres ==
 +
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 &pi; in those cases.
 
Note that the Gamma function is required for odd dimensions and that its value cancels out the apparent fractional power of &pi; in those cases.
  
 +
== Gaussian Function ==
 +
[[Image:Gaussian.jpg|400px|right]]
 +
 +
<math>f(x) = a e^{- { \frac{(x-b)^2 }{ 2 c^2} } }</math>
 +
 +
for some [[real number|real]] constants ''a'', ''b'', ''c'' > 0
  
 +
*[[W:Gaussian function]]
 +
*[http://fooplot.com/index.php?&type0=0&type1=0&type2=0&type3=0&type4=0&y0=exp%28-x%5E2%29&y1=exp%28-x%5E4%29&y2=exp%28-x%5E6%29&y3=exp%28-x%5E10000%29&xmin=-2.5260870399999997&xmax=2.551352960000001&ymin=-1.1235306666666665&ymax=2.1602026666666676 FooPlot link to live graph]
  
 +
== Pi ==
 +
*[[w:Bailey-Borwein-Plouffe formula]] ''- provides a [[w:spigot algorithm|spigot algorithm]] for the computation of the n<sup>th</sup> binary digit of pi (symbol: π) using base 16 math.''
  
----
+
== See also ==
;See also
+
*[[FFT]]
 
*[[w:Exponential function|Exponential function]]
 
*[[w:Exponential function|Exponential function]]
 
*[[w:E (mathematical constant)|E (mathematical constant)]]
 
*[[w:E (mathematical constant)|E (mathematical constant)]]
 
*[[w:Euler's identity|Euler's identity]]
 
*[[w:Euler's identity|Euler's identity]]
 
*[[w:Gamma function|Gamma function]]
 
*[[w:Gamma function|Gamma function]]
 +
*[http://www.math.utah.edu/~palais/pi.html Pi is wrong!]
 +
*[https://www.youtube.com/watch?v=sj8Sg8qnjOg About Phi being the most irrational number]
 +
[[Category:Maths]][[Category:nad]][[Category:philosophy]]

Latest revision as of 22:39, 8 June 2021

A page for e, i, π, and φ
Approximation of ex with first four Maclaurin series polynomials

The role that the exponential function (ex) plays is extremely significant to an understanding of the laws of nature and harmonic organisation. Mathematically (ex) can be expressed several ways. A common form is the Maclaurin series expansion of;

[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]

Note that;

[math]e^1 = \sum_{n=0}^{\infty}{1^n \over n!} = 1 + 1 + {1^2 \over 2!} + {1^3 \over 3!} + {1^4 \over 4!} + \cdots[/math]

and

[math]\frac{1}{e} = \sum_{n=0}^{\infty} (-1)^n\frac{1}{n!} = 1-\frac{1}{1!}+\frac{1}{2!}-\frac{1}{3!}+\cdots[/math]

where

[math] e^2 = e^{(1+1)} = e^1 \times e^1 [/math].

or

[math] \sum_{n=0}^{\infty}{2^n \over n!} = \sum_{n=0}^{\infty}{1^n \over n!} \times \sum_{m=0}^{\infty}{1^m \over m!}[/math].

Expansion of the right hand side terms is an outer product, taking the summation of these terms equals e2.

This formula needs to be checked and converted to LaTeX

As an infinite sum ex is;

[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]

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:

# Returns e^x from e( x, iterations )
sub e {
	$x = shift;
	$e = $top = $bot = 1;
	for $n ( 1 .. shift ) {
		$top = $top * $x;
		$bot = $bot * $n;
		$e = $e + $top / $bot;
	}
	return $e;
}

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.

# Returns e^x from e( x, iterations )
sub e {
	$x = $y = shift;
	$top = $bot = 1;
	for $n ( 1 .. shift ) {
		$y = $y + $x;
		$top = $top * $n + $y;
		$bot = $bot * $n;
	}
	return $top / $bot;
}

Here's a snipit in C which uses the taylor series of cosine to make Φ from π

// Calculate Phi from pi
double Phi = 0;
int n, i = -1;
for (n = 0; n < iterations; n += 2) {
	Phi += pow(PI/5,n) / factorial(n) * (i = -i);
}
Phi *= 2;

The unit circle

The nth roots of unity, or de Moivre numbers, are all the complex numbers which yield 1 when raised to a given power n. It can be shown that they are located on the unit circle of the complex plane and that in that plane they form the vertices of a n-sided regular polygon with one vertex on 1.

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 series expansion

Taylor's theorem expresses a function f(x) as a power series in x, basically because the nth derivative of xn is n!.

Why is it that when we reduce the function down to degree-1 through differentiation it ends up as n!? --Nad 17:16, 26 Oct 2006 (NZDT)
Because each derivative multiplies by the power and reduces the power by one. --Nad 12:12, 12 July 2007 (NZST)
So maybe the n! denominator in talor-series of ex should be thought of as the nth derivative rather than as combinatorial. --Nad 12:16, 12 July 2007 (NZST)

Actually, the Taylor series article explains it - the theorem in general for a function f(a) is,

[math]\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!} (x-a)^{n}\,,[/math]

where f (n)(a) denotes the nth derivative of f at the point a; the zeroth derivative of f is defined to be f itself and [math](x-a)^0[/math] is defined to be 1. But all the derivatives are the still ex for the exponential function, and we're using a=0, so the top term here is always equal to 1, leaving only xn on the top. --Nad 18:22, 12 July 2007 (NZST)

Spheres

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.

Gaussian Function

Gaussian.jpg

[math]f(x) = a e^{- { \frac{(x-b)^2 }{ 2 c^2} } }[/math]

for some real constants a, b, c > 0

Pi

See also