The Bessel Functions

Solutions to the Differential Equations

The Bessel functions are found in the solutions to Bessel's Differential Equation.

`x^2y''+xy'+(x^2-v^2)y=0`

As a second order differential equation, there are two linearly independant solutions. These solutions involve two Bessel functions, The First Kind, Jv, and The Second Kind, Yv. Thus, this first Bessel Diffferential Equation has the solution,

`y(x)=c_1J_v(x)+c_2Y_v(x)`
where the argument, x, is complex valued

A second differential equation, The Modified Bessel equation, have solutions involving modified Bessel functions,

`x^2y''+xy'-(x^2+v^2)y=0`

These two modified functions include The Modified First Kind, Iv, and The Modified Second Kind, Kv. This Modified Bessel Differential equation has the solution,

`y(x)=c_1I_v(x)+c_2K_v(x)`
where the argument, x, is purely imaginary

Linear Combinations

There are two additional complex valued functions defined as linear combinations of the Bessels functions of the First and Second kind, these are the Bessel Functions of the Third Kind (also known as the Hankel Functions),

`H_v^((1))=J_v(x)+iY_v(x)`
and
`H_v^((2))=J_v(x)-iY_v(x)`

Summary of the Bessel Functions

`J_v(x)` The First Kind
`Y_v(x)` The Second Kind
`I_v(x)` The Modified First Kind
`K_v(x)` The Modified Second Kind
`H_v^((1))(x)` The Third Kind
`H_v^((2))(x)` The Third Kind

Relationships between the Bessel Functions

`I_v(z)=i^(-v)J_v(iz)`

`K_v(z)=pi/(2sinpiv) [I_(-v)(z)-I_v(z)]`

`Y_v(z)=cot(piv)J_v(z)-csc(piv)J_(-v)(z)`

 

Properties of `J_v(x)`

Symmetry  

`J_v(-x)=J_(-v)(x)=(-1)^vJ_v(x)`
Recurrence   `2J'_v(x)=J_(v-1)(x)-J_(v+1)(x)`
Recurrence   `(2v)/x J_v(x)=J_(v+1)(x)+J_(v-1)(x)`
Derivative   `J_v(ax+c)'=a[(vJ_v(ax+c))/(ax+c)-J_(v+1)(ax+c)]`
Derivative   `d/(dx)(x^vJ_v(x))=x^vJ_(v-1)(x)`
Integral   `int x^(v+1)J_v(x)dx=x^(v+1)J_(v+1)(x)`
Integral   `int x^(1-v)J_v(x)dx=-x^(1-v)J_(v+1)(x)`

Properties of `Y_v(x)`

`Y_v(-x)=-Y_(-v)(x)=(-1)^vY_v(x)`

`Y_(v-1)(x)+Y_(v+1)(x)=(2v)/xY_v(x)`

`2Y'_v(x)=Y_(v-1)(x)-Y_(v+1)(x)`

`Y_v(ax+c)'=a[(vY_v(ax+c))/(ax+c)-Y_(v+1)(ax+c)]`

`d/(dx)(x^vY_v(x))=x^vY_(v-1)(x)`

`int x^(v+1)Y_v(x)dx=x^(v+1)Y_(v+1)(x)`

`int x^(1-v)Y_v(x)=-x^(1-v)Y_(v+1)(x)`

Properties of `I_v(x)`

`I_(-v)=(-1)^vI_v(-x)=I_v(x)`

`I_(v-1)(x)-I_(v+1)(x)=(2v)/xI_v(x)`

`2I'_v(x)=I_(v-1)(x)+I_(v+1)(x)`

`I_v(ax+c)'=a[(vI_v(ax+c))/(ax+c)+I_(v+1)(ax+c)]`

`d/(dx)(x^vI_v(x))=x^vI_(v-1)(x)`

`int x^(v+1)I_v(x)dx=x^(v+1)I_(v+1)(x)`

`int x^(1-v)I_v(x)dx=-x^(1-v)I_(v+1)(x)`

Properties of `K_v(x)`

`K_(-v)(x)=(-1)^vK_v(-x)=K_v(x)`

`K_(v+1)(x)-K_(v-1)(x)=(2v)/xK_v(x)`

`K_(v-1)(x)-Y_(v+1)(x)=2Y'_v(x)`

`K_v(ax+c)'=a[(vK_v(ax+c))/(ax+c)-K_(v+1)(ax+c)]`

`d/(dx)(x^vK_v(x))=-x^vK_(v-1)(x)`

`int x^(v+1)K_v(x)dx=-x^(v+1)K_(v+1)(x)`

`int x^(1-v)K_v(x)dx=x^(1-v)K_(v+1)(x)`

Integral Representations

`J_v(x)=1/pi int_0^pi cos(xsint-vt)dt-(sin(piv))/pi int_0^oo exp(-xsinh(t)-vt)dt`

`Y_v(x)=1/pi int_0^pi sin(xsint-vt)dt-1/pi int_0^oo exp(-xsinh(t)) (exp(vt)+cos(piv)exp(-vt))dt`

`I_v(x)=1/pi int_0^pi exp(x cost) cos(vt)dt - (sin(piv))/pi int_0^oo exp(-x cosh(t)-vt)dt`

`K_v(x)=int_0^oo exp(-x cosh(t))cosh(vt)dt`

Series Expansions

for real-valued order,

`J_v(x)=sum_(n=0)^oo (-1)^n / (n!(n+v)!) (x/2)^(2n+v)`

for complex-valued order,

`J_v(x)=sum_(n=0)^oo (-1)^n/ (n! Gamma(n+v+1)) (x/2)^(2n+v)`

`J_(-v)(x)=(-1)^v J_v(x)`

`Y_v(x)=(J_v(x)cos(vpi)-J_(-v)(x))/sin(vpi)`

`I_v(x)=(x/2)^v sum_(k=0)^oo 1/(Gamma(k+1)Gamma(v+1+k)) (x/2)^(2k)`

`K_v(x)=pi/2 (I_(-v)(x)-I_v(x))/sin(vpi)`

Other Useful Differential Forms

ODE   Solution
`x^2y''+xy'+(x^2-v^2)y=0`   `y=c_1J_v(x)+c_2Y_v(x)`
`xy''+y'+xy=0`   `y=c_1J_0(x)+c_2Y_0(x)`
`xy''+y'+by=0`   `y=c_1J_0(2sqrt(bx))+c_2Y_0(2sqrt(bx))`
`xy''+ay'+by=0`   `y=x^(1/2-a/2)[c_1J_(a-1)(2sqrt(bx))+c_2Y_(a-1)(2sqrt(bx))]`
`xy''+ay'+xby=0`   `y=x^(1/2-a/2)[c_1J_(a/2-1/2)(xsqrt(b))+c_2Y_(a/2-1/2)(xsqrt(b))]`

Differentiation

We can differentiate the Bessel functions using the fundamental derivative formula (above), and when necessary, using the chain rule. For example, say we want to differentiate the function:

`y=J_0(sqrt(kt))`

We begin by stating this is a problem involving a chain of functions. Thus by using u-substitution, we define, `u=sqrt(kt)`, such that `(du)/(dt)=sqrt(k) (d)/(dt)sqrt(t)`. Thus,

`(dy)/(dt)=(dJ_0(sqrt(kt)))/(dt)=(dJ_0(u))/(du) (du)/(dt)`

Next we evaluate the derivative `u'(t)`,

`(du)/(dt)=sqrt(k) (d)/(dt)sqrt(t)=sqrt(k) (d)/(dt) t^(1/2)=sqrt(k)/2 t^(-1/2)=sqrt(k)/(2sqrt(t))`

Then we evaluate the derivative `J_0'(u)`, which we do by using the fundamental derivative of the Bessel Jv function, cf.

`2J'_v(x)=J_(v-1)(x)-J_(v+1)(x)`

where v=0

Thus,

`2J'_0(x)=J_(-1)(x)-J_(1)(x)`

Using the symmetry relation,

`J_v(-x)=J_(-v)(x)=(-1)^vJ_v(x)`

We have,

`J'_0(x)=1/2[-J_(1)(x)-J_(1)(x)]=-J_1(x)`

Thus our derivative becomes,

`(dy)/(dt)=(dJ_0(sqrt(kt)))/(dt)=(dJ_0(u))/(du) (du)/(dt)=-J_1(u) sqrt(k)/(2sqrt(t))`

Reversing the substitution for u, we arrive at,

`(dy)/(dt)=-J_1(sqrt(kt)) sqrt(k)/(2sqrt(t))`

Algorithms for Evaluation

To evaluate the Bessel functions, we can use the series expansion for `J_v(x)`,

`J_v(x)=sum_(n=0)^oo (-1)^n / (n!(n+v)!) (x/2)^(2n+v)`

On this website, we evaluate the Bessel-functions using JScript, since this is cross-browser compliant. We use the code below,

function BesselJ(v,z,m)
//The BesselJ Function in JScript
//m is the number of terms to include
{
  var rval=0;
  for (var n = 0; n <= m; n++){
  rval=rval+Math.pow(-1,n)/(sFact(n)*sFact(n+v))*Math.pow((z/2),2*n+v);
}
return rval;
}

Excel has functionality built-in to evaluate the Bessel-functions.

References