Fourier series solution for Laplace equation in polar coordinates results in Power::infy and Infinity::indet

Question

Definition of function f

f[θ_] = 1 + Cos[2 θ] + Sin[3 θ]
Plot[f[θ], {θ, 0, 2 π}]

Computation of the Fourier coefficients

a[0] = 1/(2 π)*Integrate[f[θ], {θ, 0, 2 π}]
a[n_] = 1/(π a^n)*Integrate[f[θ]*Cos[n*θ], {θ, 0, 2 π}]
 Simplify[a[n], Element[n, Integers]]
 b[n_] = 1/(π a^n)*Integrate[f[θ]*Sin[n*θ], {θ, 0, 2 π}]
 Simplify[b[n], Element[n, Integers]]

Construction of the exact solution resulting in the following warning:

  Nt = 10
  u[ρ_, θ_] = Sum[ρ^k*b[k]*Sin[(k)*θ], {k, 0, Nt}]

Power::infy: Infinite expression 1/0 encountered.

Infinity::indet: Indeterminate expression 0 ComplexInfinity encountered.

What must I change to my code to receive the first ten coefficients of $a_n$ and $b_n$?

Answer 1

You've encountered a known issue of Integrate elaborated in the following post:

Usage of Assuming for Integration

For your specific problem one possible work-around is to avoid symbolic integration for general n by changing = to :=:

f[θ_] = 1 + Cos[2 θ] + Sin[3 θ];
a[0] = 1/(2 π) Integrate[f[θ], {θ, 0, 2 π}];
a[n_] := 1/(π a^n) Integrate[f[θ] Cos[n θ], {θ, 0, 2 π}]
b[n_] := 1/(π a^n) Integrate[f[θ] Sin[n θ], {θ, 0, 2 π}]
Nt = 10;
u[ρ, θ] = Sum[ρ^k (b[k] Sin[k θ] + a[k] Cos[k θ]), {k, 0, Nt}]
(* 1 + (ρ^2 Cos[2 θ])/a^2 + (ρ^3 Sin[3 θ])/a^3 *)

Another work-around is to use Limit to keep away from the removable singularity:

f[θ_] = 1 + Cos[2 θ] + Sin[3 θ];
a[0] = Integrate[f[θ], {θ, 0, 2 Pi}]/(2 π);
a[n_] = Integrate[f[θ] Cos[n θ], {θ, 0, 2 Pi}]/(π a^n);
b[n_] = Integrate[f[θ] Sin[n θ], {θ, 0, 2 Pi}]/(π a^n);
Nt = 10;
u[ρ, θ] = a[0] + Sum[Limit[ρ^k (b[k] Sin[k θ] + a[k] Cos[k θ]), k -> i], {i, Nt}]
(* 1 + (ρ^2 Cos[2 θ])/a^2 + (ρ^3 Sin[3 θ])/a^3 *)