Manuals/calci/RIEMANNZETA - Revision history

Swapna at 20:22, 18 July 2015

2015-07-18T20:22:29Z

Devika: /* See Also */

2015-05-12T17:44:03Z

Devika at 17:41, 12 May 2015

2015-05-12T17:41:45Z

Devika at 08:06, 25 July 2014

2014-07-25T08:06:27Z

Abin at 07:25, 25 July 2014

2014-07-25T07:25:35Z

Devika: /* Description */

2014-07-25T06:13:38Z

Description

Devika at 06:12, 25 July 2014

2014-07-25T06:12:52Z

Devika: /* Description */

2014-07-25T06:10:56Z

Description

Devika: /* Description */

2014-07-25T05:56:02Z

Description

Devika at 05:44, 25 July 2014

2014-07-25T05:44:19Z

@@ Line 33: / Line 33: @@
 #=RIEMANNZETA(4) = π^4/90
 #=RIEMANNZETA(-25)= -657931/12.
 ==See Also==

@@ Line 38: / Line 38: @@
 *[[Manuals/calci/MOODSMEDIANTEST| MOODSMEDIANTEST]]
 *[[Manuals/calci/FRIEDMANTEST| FRIEDMANTEST]]
-*[[Manuals/calci/KSNORMALTEST| KSNORMALTEST]]
+*[[Manuals/calci/KSTESTNORMAL| KSTESTNORMAL]]
 ==References==
 *[http://en.wikipedia.org/wiki/Riemann_zeta_function Riemann Zeta Function]

← Older revision		Revision as of 17:41, 12 May 2015
Line 33:		Line 33:
	#=RIEMANNZETA(4) = π^4/90		#=RIEMANNZETA(4) = π^4/90
	#=RIEMANNZETA(-25)= -657931/12.		#=RIEMANNZETA(-25)= -657931/12.
		+
		+	==See Also==
		+	*[[Manuals/calci/LEVENESTEST\| LEVENESTEST]]
		+	*[[Manuals/calci/MOODSMEDIANTEST\| MOODSMEDIANTEST]]
		+	*[[Manuals/calci/FRIEDMANTEST\| FRIEDMANTEST]]
		+	*[[Manuals/calci/KSNORMALTEST\| KSNORMALTEST]]
		+
		+	==References==
		+	*[http://en.wikipedia.org/wiki/Riemann_zeta_function Riemann Zeta Function]

@@ Line 27: / Line 27: @@
 **when s=2, then ζ (2) derivation is Basel problem.
 **when s=3, then ζ (3) derivation is Apery's constant.
-**When s=4, then ζ (4) derivation is Planck's law
+**When s=4, then ζ (4) derivation is Planck's law.

@@ Line 1: / Line 1: @@
 <div style="font-size:30px">'''RIEMANNZETA(s)'''</div><br/>
-*<math>s</math> is the value from 0 to infinity.
+*<math>s</math> is the value from <math>0</math> to <math>infinity</math>.
 ==Description==

@@ Line 15: / Line 15: @@
 *It is also  a function of a complex variable s that analytically continues the sum of the infinite series , which converges when the real part of s is greater than 1.
 *It is defined by :<math>\zeta(s)=\sum_{n=1}^\infty n^{-s}= \frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+.....</math> where <math>\sigma</math> =Real part of s>1.
-*We can define this by integral also:<math>zeta(s)=\frac{1}{\Gamma(s)} \int\limits_{0}^{\infty}\frac{x^{s-1}}{e^{x-1}}\,dx </math>
+*We can define this by integral also:<math>\zeta(s)=\frac{1}{\Gamma(s)} \int\limits_{0}^{\infty}\frac{x^{s-1}}{e^{x-1}}\,dx </math>
 *Riemann zeta function is a meromorphic function on the whole complex s-plane, which is holomorphic everywhere except for a simple pole at s = 1 with residue 1.
 *For  special values:

@@ Line 7: / Line 7: @@
 *This function is useful in number theory for the investigating properties of prime numbers.
 *It is denoted by <math>\zeta(s)</math>.
-*This function is defined as the infinite series ζ(s)=1+2^-s+3^-s+.....
+*This function is defined as the infinite series <math>\zeta(s)=1+2^{-s}+3^{-s}+.....</math>.
 *When the value of s=1,then this series is called the harmonic series.
 *When it is increase without any bound or limit, then its sum is infinite.
 *When the value of s is larger than 1,the  the series converges to a finite number as successive terms are added.
 *The riemann zeta function is defined for Complex numbers also.
-*So  ζ(s) is a function of a complex variable s = σ + it.
+*So  <math>\zeta(s)</math> is a function of a complex variable <math>s = \sigma + it</math>,where <math>\sigma</math> and t are real numbers.i is the imaginary unit.
-*It is a function of a complex variable s that analytically continues the sum of the infinite series , which converges when the real part of s is greater than 1.
+*It is also  a function of a complex variable s that analytically continues the sum of the infinite series , which converges when the real part of s is greater than 1.
-*It is defined by :ζ(s)=summation n= 1 to infinity n^-s= 1/1^s+1/2^s+1/3^s+..... where σ =Real part of s>1.
+*It is defined by :<math>\zeta(s)=\sum_{n=1}^\infty n^{-s}= \frac{1}{1^s}+\frac{1}{2^s}+\frac{1}{3^s}+.....</math> where <math>\sigma</math> =Real part of s>1.
-*We can define this by integral also:ζ(s)=1/gamma (s)integral 0 to infinity x^(s-1)/e^x-1 dx.
+*We can define this by integral also:<math>zeta(s)=\frac{1}{\Gamma(s)}\int\limits_{0}^{\infty} \frac{x^{s-1}{e^{x-1}\,dx </math>
 *Riemann zeta function is a meromorphic function on the whole complex s-plane, which is holomorphic everywhere except for a simple pole at s = 1 with residue 1.
 *(For  special values:(i)Any positive integer 2n.for n ≥ 1)so in particular ζ vanishes at the negative even integers because Bm = 0 for all odd m other than 1.

← Older revision		Revision as of 05:44, 25 July 2014
Line 1:		Line 1:
−	<div style="font-size:30px">'''RIEMANNZETA'''</div><br/>	+	<div style="font-size:30px">'''RIEMANNZETA(s)'''</div><br/>
		+	*<math>s</math> is the value from 0 10 infinity.
		+
		+	==Description==
		+	*This function gives the result for the function of Riemann-Zeta function.
		+	*It is also known as Euler-Riemann Zeta function.
		+	*This function is useful in number theory for the investigating properties of prime numbers.
		+	*It is denoted by <math>\zeta(s)</math>.
		+	*This function is defined as the infinite series ζ(s)=1+2^-s+3^-s+.....
		+	*When the value of s=1,then this series is called the harmonic series.
		+	*When it is increase without any bound or limit, then its sum is infinite.
		+	*When the value of s is larger than 1,the the series converges to a finite number as successive terms are added.
		+	*The riemann zeta function is defined for Complex numbers also.
		+	*So ζ(s) is a function of a complex variable s = σ + it.
		+	*It is a function of a complex variable s that analytically continues the sum of the infinite series , which converges when the real part of s is greater than 1.
		+	*It is defined by :ζ(s)=summation n= 1 to infinity n^-s= 1/1^s+1/2^s+1/3^s+..... where σ =Real part of s>1.
		+	*We can define this by integral also:ζ(s)=1/gamma (s)integral 0 to infinity x^(s-1)/e^x-1 dx.
		+	*Riemann zeta function is a meromorphic function on the whole complex s-plane, which is holomorphic everywhere except for a simple pole at s = 1 with residue 1.
		+	*(For special values:(i)Any positive integer 2n.for n ≥ 1)so in particular ζ vanishes at the negative even integers because Bm = 0 for all odd m other than 1.
		+	*For odd positive integers, no such simple expression is known.
		+	**When s=1,then ζ (1) is Harmonic series.
		+	**when s=2, then ζ (2) derivation is Basel problem.
		+	**when s=3, then ζ (3) derivation is Apery's constant.
		+	**When s=4, then ζ (4) derivation is Planck's law