An Analytical Solution To A Special Case Of The Weber-Schafheitlin Integral
Special cases of Weber-Schafheitlin type integrals are evaluated analytically.
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arXiv subjects
R. Mehrem
Publications and source records attributed to R. Mehrem.
Non-Local Interactions For the Deuteron Using Spherical Bessel Functions
Non-local interactions are assumed for the deuteron with form factors $g_C(k)= j_0(b_1k)\,j_0(b_2k)$ for the central part responsible for the S-state and $g_T(k)= j_1(b_1k)\,j_1(b_2k)$ for the tensor part responsible for the D-state, where $b_1$ and $b_2$ are range parameters for the proton and neutron, respectively. The analytically obtained wavefunctions in coordinate space have different forms in three different regions. The inner most region is between $r=0$ and $r=b_2-b_1$ (assuming $b_2>b_1$), followed by a region between $r=b_2-b_1$ and $b_1+b_2$ and finally the region $r>b_1+b_2$. The resulting wavefunctions and their derivatives are found to be continuous at the boundaries. The ensuing calculations are simplified by setting $b_1=b_2=b$. Good agreement is obtained for the quadrupole moment. Neutron-proton scattering calculations will follow in the next communication.
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An Analytical Evaluation For The Integral Of Two Spherical Bessel Functions With An Additional Exponential And Polynomial Factor
The integrals $\threej{\Llo}{\Llt}{\Llth} {0}{0}{0}\,\int_0^\infty \,r^{\Llth+1}\,e^{-αr}\,j_\Llo(k_1r)\, j_\Llt(k_2r)\,dr$ and $\threej{\Llo}{\Llt}{\Llth} {0}{0}{0}\,\int_0^\infty \,r^{\Llth+2}\,e^{-αr}\,j_\Llo(k_1r)\, j_\Llt(k_2r)\,dr$ are evaluated analytically. The result is a finite sum over the associated Legendre function of the second kind.
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Analytical Evaluation Of An Infinite Integral Over Four Spherical Bessel Functions
An infinite integral over four spherical Bessel functions is analytically evaluated for the special case when the arguments k_3=k_1 and k_4=k_2
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A Generalisation For The Infinite Integral Over Three Spherical Bessel Functions
A new formula is derived that generalises an earlier result for the infinite integral over three spherical Bessel functions. The analytical result involves a finite sum over associated Legendre functions, $P_l^m(x)$, of degree $l$ and order $m$. The sum allows for values of $|m|$ that are greater than $l$. A generalisation for the associated Legendre functions to allow for any rational $m$ for a specific $l$ is also shown
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