The pdf of the wrapped Lévy distribution is
f W L ( θ ; μ , c ) = ∑ n = − ∞ ∞ c 2 π e − c / 2 ( θ + 2 π n − μ ) ( θ + 2 π n − μ ) 3 / 2 where the value of the summand is taken to be zero when θ + 2 π n − μ ≤ 0 , c is the scale factor and μ is the location parameter. Expressing the above pdf in terms of the characteristic function of the Lévy distribution yields:
f W L ( θ ; μ , c ) = 1 2 π ∑ n = − ∞ ∞ e − i n ( θ − μ ) − c | n | ( 1 − i sgn n ) = 1 2 π ( 1 + 2 ∑ n = 1 ∞ e − c n cos ( n ( θ − μ ) − c n ) ) In terms of the circular variable z = e i θ the circular moments of the wrapped Lévy distribution are the characteristic function of the Lévy distribution evaluated at integer arguments:
⟨ z n ⟩ = ∫ Γ e i n θ f W L ( θ ; μ , c ) d θ = e i n μ − c | n | ( 1 − i sgn ( n ) ) . where Γ is some interval of length 2 π . The first moment is then the expectation value of z, also known as the mean resultant, or mean resultant vector:
⟨ z ⟩ = e i μ − c ( 1 − i ) The mean angle is
θ μ = A r g ⟨ z ⟩ = μ + c and the length of the mean resultant is
R = | ⟨ z ⟩ | = e − c 