Difference between revisions of "Manuals/calci/BESSELI"

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<div style="font-size:30px">'''BESSELI(x,n)'''</div><br/>
 
<div style="font-size:30px">'''BESSELI(x,n)'''</div><br/>
 
*<math>x</math> is the value to evaluate the function
 
*<math>x</math> is the value to evaluate the function
*<math>n</math> is an integer which is the order of the Bessel function
+
*<math>n</math> is an integer which is the order of the Bessel function.
 +
**BESSELI(), returns the modified Bessel Function In(x).
 +
 
 
==Description==
 
==Description==
 
*This function gives the value of the modified Bessel function.
 
*This function gives the value of the modified Bessel function.
 
*Bessel functions is also called Cylinder Functions because they appear in the solution to Laplace's equation in cylindrical coordinates.
 
*Bessel functions is also called Cylinder Functions because they appear in the solution to Laplace's equation in cylindrical coordinates.
*Bessel's Differential Equation is defined as: <math>x^2 \frac{d^2 y}{dx^2} + x\frac{dy}{dx} + (x^2 - \alpha^2)y =0</math>
+
*Bessel's Differential Equation is defined as:
 +
<math>x^2 \frac{d^2 y}{dx^2} + x\frac{dy}{dx} + (x^2 - \alpha^2)y =0</math>  
 
where <math>\alpha</math> is the arbitrary complex number.
 
where <math>\alpha</math> is the arbitrary complex number.
 
*But in most of the cases α is the non-negative real number.
 
*But in most of the cases α is the non-negative real number.
 
*The solutions of this equation are called Bessel Functions of order <math>n</math>.
 
*The solutions of this equation are called Bessel Functions of order <math>n</math>.
*Bessel functions of the first kind, denoted as <math>Jn(x)</math>.  
+
*Bessel functions of the first kind, denoted as <math>J_n(x)</math>.  
*The <math>n^th</math> order modified Bessel function of the variable <math>x</math> is: <math>In(x)=i^{-n}Jn(ix)}</math> ,where <math>Jn(x)=\sum_k=0 \infinity){(-1)^k(x/2)^n+2k}/k!gamma(n+k+1).
+
*The <math>n^{th}</math> order modified Bessel function of the variable <math>x</math> is:  
*This function will give the result as error when 1.x or n is non numeric2. n<0, because n is the order of the function.
+
<math>I_n(x)=i^{-n}J_n(ix)</math>,  
 +
where :
 +
<math>J_n(x)=\sum_{k=0}^\infty \frac{(-1)^k*(\frac{x}{2})^{n+2k} }{k!\Gamma(n+k+1)}</math>
 +
*This function will give the result as error when:
 +
1.<math>x</math> or <math>n</math> is non numeric
 +
2.<math>n<0</math>, because <math>n</math> is the order of the function.
 +
 
 +
==ZOS==
 +
*The syntax is to calculate BESSELI IN ZOS is <math>BESSELI(x,n)</math>.
 +
**<math>x</math> is the value to evaluate the function
 +
**<math>n</math> is an integer which is the order of the Bessel function.
 +
*For e.g.,BESSELI(0.25..0.7..0.1,42)
  
 
==Examples==
 
==Examples==
  
#BESSELI(3,2)=2.245212431(Excel) this is the n th derivative(In(x))=3.9533702171(Calci)this is the 1st derivative(I1(x))
+
#BESSELI(3,2) = 2.245212431 this is the <math>2^{nd}</math> derivative of <math>I_n(x)</math>.
#BESSELI(5,1)=24.33564185
+
#BESSELI(5,1) = 24.33564185
#BESSELI(6,0)=67.23440724(Excel)  I0(x)61.3419369373(CALCI) I1(x)
+
#BESSELI(6,0) = 67.23440724
#BESSELI(-2,1)=0.688948449(Excel) =-1.5906368573(CALCI)
+
#BESSELI(-2,1) = -1.59063685
#BESSELI(2,-1)=NAN ,because n<0.
+
#BESSELI(2,-1) = #N/A (ORDER OF FUNCTION < 0).
 +
 
 +
==Related Videos==
 +
 
 +
{{#ev:youtube|__fdGscBZjI|280|center|BESSEL Equation}}
  
 
==See Also==
 
==See Also==
Line 27: Line 45:
  
 
==References==
 
==References==
[http://en.wikipedia.org/wiki/Bessel_function | Bessel Function]
+
[http://en.wikipedia.org/wiki/Bessel_function Bessel Function]
 +
 
 +
 
 +
 
 +
*[[Z_API_Functions | List of Main Z Functions]]
 +
 
 +
*[[ Z3 |  Z3 home ]]

Latest revision as of 04:23, 18 November 2020

BESSELI(x,n)


  • is the value to evaluate the function
  • is an integer which is the order of the Bessel function.
    • BESSELI(), returns the modified Bessel Function In(x).

Description

  • This function gives the value of the modified Bessel function.
  • Bessel functions is also called Cylinder Functions because they appear in the solution to Laplace's equation in cylindrical coordinates.
  • Bessel's Differential Equation is defined as:

where is the arbitrary complex number.

  • But in most of the cases α is the non-negative real number.
  • The solutions of this equation are called Bessel Functions of order .
  • Bessel functions of the first kind, denoted as .
  • The order modified Bessel function of the variable is:

, where :

  • This function will give the result as error when:
1. or  is non numeric
2., because  is the order of the function.

ZOS

  • The syntax is to calculate BESSELI IN ZOS is .
    • is the value to evaluate the function
    • is an integer which is the order of the Bessel function.
  • For e.g.,BESSELI(0.25..0.7..0.1,42)

Examples

  1. BESSELI(3,2) = 2.245212431 this is the derivative of .
  2. BESSELI(5,1) = 24.33564185
  3. BESSELI(6,0) = 67.23440724
  4. BESSELI(-2,1) = -1.59063685
  5. BESSELI(2,-1) = #N/A (ORDER OF FUNCTION < 0).

Related Videos

BESSEL Equation

See Also

References

Bessel Function