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Bessel functions of the first kind, denoted as J α (x), are solutions of Bessel's differential equation. For integer or positive α, Bessel functions of the first kind are finite at the origin (x = 0); while for negative non-integer α, Bessel functions of the first kind diverge as x approaches zero.
However, the modified Bessel functions of the second kind () also satisfy the same recurrence relation = + + (). The first solution decreases rapidly with . The second solution increases rapidly with . Miller's algorithm provides a numerically stable procedure to obtain the decreasing solution.
I 0 is the zeroth-order modified Bessel function of the first kind, L is the window duration, and; α is a non-negative real number that determines the shape of the window. In the frequency domain, it determines the trade-off between main-lobe width and side lobe level, which is a central decision in window design.
In the free boundary conditions case, the Hamiltonian is = [ + + ()] therefore the partition function factorizes under the change of coordinates = ′ + This gives = = = ′ ′ = [′ ′] = (()) where is the modified Bessel function of the first kind. The partition function can be used to find several important ...
The probability density function is (,) = ((+)) (),where I 0 (z) is the modified Bessel function of the first kind with order zero.. In the context of Rician fading, the distribution is often also rewritten using the Shape Parameter =, defined as the ratio of the power contributions by line-of-sight path to the remaining multipaths, and the Scale parameter = +, defined as the total power ...
where and , > and is the modified Bessel function of first kind of order . If b > 0 {\displaystyle b>0} , the integral converges for any ν {\displaystyle \nu } . The Marcum Q-function occurs as a complementary cumulative distribution function for noncentral chi , noncentral chi-squared , and Rice distributions .
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The von Mises probability density function for the angle x is given by: [2] (,) = ( ()) ()where I 0 is the modified Bessel function of the first kind of order 0, with this scaling constant chosen so that the distribution sums to unity: () = ().