Laplace equation in cylindrical coordinates

1 Laplace Equation in Cylindrical Coordinates

Solutions to the Laplace equation in cylindrical coordinates have wide applicability from fluid mechanics to electrostatics. Applying the method of separation of variablesMathworldPlanetmath to Laplace’s partial differential equationMathworldPlanetmath and then enumerating the various forms of solutions will lay down a foundation for solving problems in this coordinate systemMathworldPlanetmath. Finally, the use of Bessel functionsDlmfMathworldPlanetmathPlanetmath in the solution reminds us why they are synonymous with the cylindrical domain.

1.1 Separation of Variables

Beginning with the LaplacianDlmfMathworld in cylindrical coordinatesMathworldPlanetmath, apply the operator to a potential function and set it equal to zero to get the Laplace equation

2Φ=1rr(rΦr)+1r22Φθ2+2Φz2=0. (1)

First expand out the terms

2Φ=1rΦr+2Φr2+1r22Φθ2+2Φz2=0. (2)

Then apply the method of separation of variables by assuming the solution is in the form


Plug this into (2) and note how we can bring out the functionsMathworldPlanetmath that are not affected by the derivatives


Divide by R(r)P(θ)Z(z) and use short hand notation to get


“Separating” the z term to the other side gives


This equation can only be satisfied for all values if both sides are equal to a constant, λ, such that

-1Zd2Zdz2=λ (3)
1RrdRdr+1Rd2Rdr2+1Pr2d2Pdθ2=λ. (4)

Before we can focus on solutions, we need to further separate (4), so multiply (4) by r2


Separate the terms


As before, set both sides to a constant, κ

-1Pd2Pdθ2=κ (5)
rRdRdr+r2Rd2Rdr2-λr2=κ. (6)

Now there are three differential equations and we know the form of these solutions. The differential equations of (3) and (5) are ordinary differential equations, while (6) is a little more complicated and we must turn to Bessel functions.

1.2 Axial Solutions (z)

Following the guidelines setup in [Etgen] for linear homogeneous differential equations, the first step in solving


is to find the roots of the characteristic polynomialMathworldPlanetmathPlanetmath


Although, one can go forward using the square rootMathworldPlanetmath, here we will introduce another constant, γ to imply the following cases. So if we want real roots, then we want to ensure a negative constant


and if we want complex roots, then we want to ensure a positive constant


Case 1: λ0 and real roots (λ=-γ2).

For every real root, there will be an exponential in the general solution. The real roots are


Therefore, the solutions for these roots are


Combining these using the principle of superposition, gives the general solution,

Zγ(z)=C1eγz+C2e-γz+C3z+C4. (7)

Case 2: λ>0 and complex roots (λ=γ2).

The roots are


and the corresponding solutions


Combining these into a general solution yields


1.3 Azimuthal Solutions (θ)

Azimuthal solutions for


are in the most general sense obtained similarly to the axial solutions with the characteristic polynomial


Using another constant, ν to ensure positive or negative constants, we get two cases.

Case 1: κ0 and real roots (κ=-ν2).

The solutions for these roots are then


Combining these for the general solution,

Pν(θ)=C1eνθ+C2e-νθ+C3θ+C4. (8)

Case 2: κ>0 and complex roots (κ=ν2).

The roots are


and the corresponding solutions


Combining these into a general solution


For the first glimpse at simplification, we will note a restriction on κ that is used when it is required that the solution be periodic to ensure P is single valued


Then we are left with either the periodic solutions that occur with complex roots or the zero case. So not only


but also ν must be an integer, i.e.

Pν(θ)=C5cos(νθ)+C6sin(νθ)+C7   ν=0,1,2, (9)

Note, that ν=0, is still a solution, but to be periodic we can only have a constant


1.4 Radial Solutions (r)

The radial solutions are the more difficult ones to understand for this problem and are solved using a power series. The two types of solutions generated based on the choices of constants from the θ and z solutions (excluding non-periodic solutions for P) leads to the Bessel functions and the modified Bessel functionsDlmfDlmfDlmf. The first step for both these cases is to transform (6) into the Bessel differential equationMathworldPlanetmath.

Case 1: λ<0 (λ=-γ2), κ>0 (κ=ν2).

Substitute γ and ν into the radial equation (6) to get

rRdRdr+r2Rd2Rdr2+γ2r2-ν2=0. (10)

Next, use the substitution


Therefore, the derivatives are


and make a special note that




Substituting these relationships into (10) gives us


Finally, multiply by R/x2 to get the Bessel differential equation

d2Rdx2+1xdRdx+(1-ν2x2)R=0. (11)

Delving into all the nuances of solving Bessel’s differential equation is beyond the scope of this article, however, the curious are directed to Watson’s in depth treatise [Watson]. Here, we will just present the results as we did for the previous differential equations. The general solution is a linear combinationMathworldPlanetmath of the Bessel function of the first kind Jν(r) and the Bessel function of the second kind Yν(r). Remebering that ν is a positive integer or zero.

Rν(r)=C1Jν(γr)+C2Yν(γr)+C3 (12)

Bessel function of the first kind:


Bessel function of the second kind (using Hankel’s formula):


For the unfortunate person who has to evaluate this function, note that when m=0, the singularity is taken care of by replacing the series in brackets by


Some solace can be found since most physical problems need to be analytic at x=0 and therefore Yν(x) breaks down at ln(0). This leads to the choice of constant C2 to be zero.

Case 2: λ>0 (λ=γ2), κ>0 (κ=ν2).

Using the previous method of substitution, we just get the change of sign

d2Rdx2+1xdRdx-(1+ν2x2)R=0. (13)

This leads to the modified Bessel functions as a solution, which are also known as the pure imaginary Bessel functions. The general solution is denoted

Rν(r)=C1Iν(γr)+C2Kν(γr)+C3 (14)

where Iν is the modified Bessel function of the first kind and Kν is the modified Bessel function of the second kind


1.5 Combined Solution

Keeping track of all the different cases and choosing the right terms for boundary conditionsMathworldPlanetmath is a daunting task when one attempts to solve Laplace’s equation. The short hand notation used in [Kusse] and [Arfken] will be presented here to help organize the choices as a reference. It is important to remember that these solutions are only for the single valued azimuth cases (κ=ν2).

Once the separate solutions are obtained, the rest is simple since our solution is separable


So we just combine the individual solutions to get the general solutions to the Laplace equation in cylindrical coordinates.

Case 1: λ<0 (λ=-γ2), κ>0 (κ=ν2).

Φ(r,θ,z)=νγ{eγze-γz{cos(νθ)sin(νθ){Jν(γr)Yν(γr) (15)

Case 2: λ>0 (λ=γ2), κ>0 (κ=ν2).

Φ(r,θ,z)=νγ{cos(γz)sin(γz){cos(νθ)sin(νθ){Iν(γr)Kν(γr) (16)

Interpreting the short hand notation is as simple as expanding terms and not forgetting the linear solutions, i.e. (γ=0) . As an example, case 1, expanded out while ignoring the linear terms would give

Φ(r,θ,z)=νγ{Aνγeγzcos(νθ)Jν(γr)+Bνγeγzcos(νθ)Yν(γr)+Cνγeγzsin(νθ)Jν(γr)+Dνγeγzsin(νθ)Yν(γr)+Eνγe-γzcos(νθ)Jν(γr)+Fνγe-γzcos(νθ)Yν(γr)+Gνγe-γzsin(νθ)Jν(γr)+Hνγe-γzsin(νθ)Yν(γr)}. (17)


  • 1 Arfken, George, Weber, Hans, Mathematical Physics. Academic Press, San Diego, 2001.
  • 2 Etgen, G., Calculus. John Wiley & Sons, New York, 1999.
  • 3 Guterman, M., Nitecki, Z., Differential Equations, 3rd Edition. Saunders College Publishing, Fort Worth, 1992.
  • 4 Jackson, J.D., Classical Electrodynamics, 2nd Edition. John Wiley & Sons, New York, 1975.
  • 5 Kusse, Bruce, Westwig, Erik, Mathematical Physics. John Wiley & Sons, New York, 1998.
  • 6 Lebedev, N., Special Functions & Their Applications. Dover Publications, New York, 1995.
  • 7 Watson, G.N., A Treatise on the Theory of Bessel Functions. Cambridge University Press, New York, 1995.
Title Laplace equation in cylindrical coordinates
Canonical name LaplaceEquationInCylindricalCoordinates
Date of creation 2013-03-22 16:24:24
Last modified on 2013-03-22 16:24:24
Owner bloftin (6104)
Last modified by bloftin (6104)
Numerical id 13
Author bloftin (6104)
Entry type Derivation
Classification msc 35J05
Related topic BesselFunction
Related topic BesselsEquation