TOPICS
Search

Hankel Transform


The Hankel transform (of order zero) is an integral transform equivalent to a two-dimensional Fourier transform with a radially symmetric integral kernel and also called the Fourier-Bessel transform. It is defined as

g(u,v)=F_r[f(r)](u,v)
(1)
=int_(-infty)^inftyint_(-infty)^inftyf(r)e^(-2pii(ux+vy))dxdy.
(2)

Let

x+iy=re^(itheta)
(3)
u+iv=qe^(iphi)
(4)

so that

x=rcostheta
(5)
y=rsintheta
(6)
r=sqrt(x^2+y^2)
(7)
u=qcosphi
(8)
v=qsinphi
(9)
q=sqrt(u^2+v^2).
(10)

Then

g(q)=int_0^inftyint_0^(2pi)f(r)e^(-2piirq(cosphicostheta+sinphisintheta))rdrdtheta
(11)
=int_0^inftyint_0^(2pi)f(r)e^(-2piirqcos(theta-phi))rdrdtheta
(12)
=int_0^inftyint_(-phi)^(2pi-phi)f(r)e^(-2piirqcostheta)rdrdtheta
(13)
=int_0^inftyint_0^(2pi)f(r)e^(-2piirqcostheta)rdrdtheta
(14)
=int_0^inftyf(r)[int_0^(2pi)e^(-2piirqcostheta)dtheta]rdr
(15)
=2piint_0^inftyf(r)J_0(2piqr)rdr,
(16)

where J_0(z) is a zeroth order Bessel function of the first kind.

Therefore, the Hankel transform pairs are

g(q)=2piint_0^inftyf(r)J_0(2piqr)rdr
(17)
f(r)=2piint_0^inftyg(q)J_0(2piqr)qdq.
(18)

A slightly differently normalized Hankel transform and its inverse are implemented in the Wolfram Language as HankelTransform[expr, r, s] and InverseHankelTransform[expr, s, r], respectively.

The following table gives Hankel transforms for a number of common functions (Bracewell 1999, p. 249). Here, J_n(x) is a Bessel function of the first kind and Pi_a(r) is a rectangle function equal to 1 for 0<=r<=a and 0 otherwise, and

M(x)=2pi[x^(-3)int_0^xJ_0(x)dx-x^(-2)J_0(x)]
(19)
=(pi^2)/(x^2)[J_1(x)H_0(x)-J_0(x)H_1(x)],
(20)

where J_n(x) is a Bessel function of the first kind, H_n(x) is a Struve function and L_n(x) is a modified Struve function.

f(r)g(q)
Pi_a(r)(aJ_1(2piaq))/q
(sin(2piar))/r(Pi(q/(2a)))/(sqrt(a^2-q^2))
1/2delta(r-a)piaJ_0(2piaq)
M(ar)aLambda(q/(2a))
e^(-pir^2)e^(-piq^2)
(a^2+r^2)^(-1/2)(e^(-2piaq))/q
(a^2+r^2)^(-3/2)(2pie^(-2piaq))/a
1/(a^2+r^2)2piK_0(2piaq)
(2a^2)/((a^2+r^2)^2)4pi^2aqK_1(2piaq)
(4a^4)/((a^2+r^2)^3)4pi^3a^2q^2K_2(2piaq)
(a^2-r^2)Pi_a(r)(a^2)/(piq^2)J_2(2piaq)
1/r1/q
e^(-ar)(2pia)/((a^2+4pi^2q^2)^(3/2))
(e^(-ar))/r(2pi)/(sqrt(a^2+4pi^2q^2))
(delta(r))/(2pir)1
r^2e^(-pir^2)(e^(-piq^2)(1-piq^2))/pi
-r^2F(r)((d^2F)/(dq^2)+1/q(dF)/(dq))=del ^2F

The Hankel transform of order n is defined by

 H_n(f(t))=int_0^inftytJ_n(phit)f(t)dt
(21)

(Bronshtein et al. 2004, p. 706).

A different kind of Hankel transform can also be defined for integer sequences (Layman 2001).


See also

Bessel Function of the First Kind, Fourier Transform, Laplace Transform

Explore with Wolfram|Alpha

References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 795, 1985.Bracewell, R. "The Hankel Transform." The Fourier Transform and Its Applications, 3rd ed. New York: McGraw-Hill, pp. 244-250, 1999.Bronshtein, I. N.; Semendyayev, K. A.; Musiol, G.; and Muehlig, H. Handbook of Mathematics, 4th ed. New York: Springer-Verlag, pp. 705-706, 2004.Layman, J. W. "The Hankel Transform and Some of Its Properties." J. Integer Sequences 4, No. 01.1.5, 2001. http://www.math.uwaterloo.ca/JIS/VOL4/LAYMAN/hankel.Oberhettinger, F. Tables of Bessel Transforms. New York: Springer-Verlag, 1972.Samko, S. G.; Kilbas, A. A.; and Marichev, O. I. Fractional Integrals and Derivatives. Yverdon, Switzerland: Gordon and Breach, p. 23, 1993.

Referenced on Wolfram|Alpha

Hankel Transform

Cite this as:

Weisstein, Eric W. "Hankel Transform." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/HankelTransform.html

Subject classifications