Substituting Eq. (2) into Eq. (3) yields an expression for the AoLP central moments involving two integrals Display Formula
$\u2329(\theta \u2212\varphi \xaf)\alpha \u232a=1\pi e\u2212d\xaf\u02dc22\u222b0\u221edp\u02dcp\u02dce\u2212p\u02dc22\u222b\varphi \xaf\u2212\pi /2\varphi \xaf+\pi /2d\theta ep\u02dc(q\xaf\u02dc\u2009cos(2\theta )+u\xaf\u02dc\u2009sin(2\theta ))(\theta \u2212\varphi \xaf)\alpha .$(13)
From Fig. 1, it can be seen that $q\xaf\u02dc=d\xaf\u02dc\u2009cos(2\varphi \xaf)$ and $u\xaf\u02dc=d\xaf\u02dc\u2009sin(2\varphi \xaf)$. Using the equality $cos(A)cos(B)+sin(A)sin(B)=cos(B\u2212A)$ and the change of variables $\beta =2\theta \u22122\varphi \xaf$ allows the angular integral from Eq. (13) to be expressed as Display Formula$12\alpha +1\u222b\u2212\pi \pi d\beta \beta \alpha ep\u02dcd\xaf\u02dc\u2009cos(\beta ).$(14)
To represent $\beta \alpha $ on the $[\u2212\pi ,\pi ]$ interval, the function can be expanded in a Fourier series Display Formula$\beta \alpha =\u2211\u2212\u221e\u221eBn\alpha ein\beta ,$(15)
where the Fourier coefficients are defined by Display Formula$Bn\alpha =12\pi \u222b\u2212\pi \pi d\beta \beta \alpha e\u2212in\beta .$(16)
This integral can be represented recursively for $n\u22600$ by integrating by parts Display Formula$Bn\alpha =(\u22121)n\pi \alpha \u22121i2n[1\u2212(\u22121)\alpha ]+\alpha inBn\alpha \u22121$(17)
and the integral can be solved for $n=0$Display Formula$B0\alpha =\pi \alpha +12\pi (\alpha +1)[1\u2212(\u22121)\alpha +1].$(18)
When $\alpha =0$ the integral in Eq. (16) simplifies to Display Formula$Bn0=12\pi \u222b\u2212\pi \pi d\beta e\u2212in\beta =\delta n0,$(19)
where $\delta $ is the Kronecker delta function. Since the Fourier coefficients have closed-form solutions, substituting Eq. (15) into Eq. (14) is helpful because it leads to an expression Display Formula$12\alpha +1\u2211n=\u2212\u221e\u221eBn\alpha \u222b\u2212\pi \pi d\beta ein\beta ep\u02dcd\xaf\u02dc\u2009cos(\beta )$(20)
that can be related to a modified Bessel function Display Formula$In(x)=12\pi \u222b\u2212\pi \pi ex\u2009cos(\theta )ein\theta d\theta .$(21)
Now the integral expression over the angle is replaced by an infinite sum involving modified Bessel functions Display Formula$2\pi 2\alpha +1\u2211n=\u2212\u221e\u221eBn\alpha In(p\u02dcd\xaf\u02dc).$(22)
By substituting these results into Eq. (13), the AoLP central moments are Display Formula$\u2329(\theta \u2212\varphi \xaf)\alpha \u232a=12\alpha e\u2212d\xaf\u02dc22\u2211n=\u2212\u221e\u221eBn\alpha \u222b0\u221edp\u02dcp\u02dce\u2212p\u02dc22In(p\u02dcd\xaf\u02dc).$(23)
Noting the recurrence relations for derivatives of modified Bessel functions Display Formula$ddxIn(x)=12In\u22121(x)+12In+1(x)$(24)
and using integration by parts from page 259 of ^{15} leads to the equality Display Formula$\u222b0\u221edp\u02dcp\u02dce\u2212p\u02dc22In(p\u02dcd\xaf\u02dc)=d\xaf\u02dc2\pi 2ed\xaf\u02dc24[In\u221212(d\xaf\u02dc24)+In+12(d\xaf\u02dc24)]\u2200\u2009\u2009n\u22600.$(25)