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fixed hankel transform formulae for specific kappa, q, n combinations. 

Former-commit-id: 6c0dae200188c83aa4fe0b2c37d1585ac192f56c
This commit is contained in:
Marek Nečada 2018-01-18 17:37:36 +02:00
parent cc417889e1
commit 0e302d53a1
17 changed files with 80 additions and 2 deletions
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4
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(-5/(Sqrt[1 + k^2/(c - I*k0)^2]*(c - I*k0)) + 10/(Sqrt[1 + k^2/(2*c - I*k0)^2]*(2*c - I*k0)) - 10/(Sqrt[1 + k^2/(3*c - I*k0)^2]*(3*c - I*k0)) + 5/(Sqrt[1 + k^2/(4*c - I*k0)^2]*(4*c - I*k0)) - 1/(Sqrt[1 + k^2/(5*c - I*k0)^2]*(5*c - I*k0)) + 1/Sqrt[k^2 - k0^2])/k0
(-5/(Sqrt(1 + Power(k,2)/Power(c - Complex(0,1)*k0,2))*(c - Complex(0,1)*k0)) + 10/(Sqrt(1 + Power(k,2)/Power(2*c - Complex(0,1)*k0,2))*(2*c - Complex(0,1)*k0)) - 10/(Sqrt(1 + Power(k,2)/Power(3*c - Complex(0,1)*k0,2))*(3*c - Complex(0,1)*k0)) + 5/(Sqrt(1 + Power(k,2)/Power(4*c - Complex(0,1)*k0,2))*(4*c - Complex(0,1)*k0)) - 1/(Sqrt(1 + Power(k,2)/Power(5*c - Complex(0,1)*k0,2))*(5*c - Complex(0,1)*k0)) + 1/Sqrt(Power(k,2) - Power(k0,2)))/k0
SeriesData[k, Infinity, {-(1/(k*k0)), k0^(-1), -k0/(2*k), k0/2, ((-15*(c - I*k0)^3*(c/k - (I*k0)/k))/8 + (15*(2*c - I*k0)^3*((2*c)/k - (I*k0)/k))/4 - (15*(3*c - I*k0)^3*((3*c)/k - (I*k0)/k))/4 + (15*(4*c - I*k0)^3*((4*c)/k - (I*k0)/k))/8 - (3*(5*c - I*k0)^3*((5*c)/k - (I*k0)/k))/8)/k0, (3*k0^3)/8, ((25*(c - I*k0)^5*(c/k - (I*k0)/k))/16 - (25*(2*c - I*k0)^5*((2*c)/k - (I*k0)/k))/8 + (25*(3*c - I*k0)^5*((3*c)/k - (I*k0)/k))/8 - (25*(4*c - I*k0)^5*((4*c)/k - (I*k0)/k))/16 + (5*(5*c - I*k0)^5*((5*c)/k - (I*k0)/k))/16)/k0, (5*k0^5)/16, ((-175*(c - I*k0)^7*(c/k - (I*k0)/k))/128 + (175*(2*c - I*k0)^7*((2*c)/k - (I*k0)/k))/64 - (175*(3*c - I*k0)^7*((3*c)/k - (I*k0)/k))/64 + (175*(4*c - I*k0)^7*((4*c)/k - (I*k0)/k))/128 - (35*(5*c - I*k0)^7*((5*c)/k - (I*k0)/k))/128)/k0, (35*k0^7)/128, ((315*(c - I*k0)^9*(c/k - (I*k0)/k))/256 - (315*(2*c - I*k0)^9*((2*c)/k - (I*k0)/k))/128 + (315*(3*c - I*k0)^9*((3*c)/k - (I*k0)/k))/128 - (315*(4*c - I*k0)^9*((4*c)/k - (I*k0)/k))/256 + (63*(5*c - I*k0)^9*((5*c)/k - (I*k0)/k))/256)/k0}, 0, 11, 1]
(-5/(Sqrt[1 + k^2/(c - I*k0)^2]*(c - I*k0)) + 10/(Sqrt[1 + k^2/(2*c - I*k0)^2]*(2*c - I*k0)) - 10/(Sqrt[1 + k^2/(3*c - I*k0)^2]*(3*c - I*k0)) + 5/(Sqrt[1 + k^2/(4*c - I*k0)^2]*(4*c - I*k0)) - 1/(Sqrt[1 + k^2/(5*c - I*k0)^2]*(5*c - I*k0)) + 1/Sqrt[k^2 - k0^2])/k0

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(-k^(-1) - 5*(k^(-1) - 1/(k*Sqrt[1 + k^2/(c - I*k0)^2])) + 10*(k^(-1) - 1/(k*Sqrt[1 + k^2/(2*c - I*k0)^2])) - 10*(k^(-1) - 1/(k*Sqrt[1 + k^2/(3*c - I*k0)^2])) + 5*(k^(-1) - 1/(k*Sqrt[1 + k^2/(4*c - I*k0)^2])) + 1/(k*Sqrt[1 + k^2/(5*c - I*k0)^2]) + (1 + (I*k0)/Sqrt[k^2 - k0^2])/k)/k0
(-(1/k) - 5*(1/k - 1/(k*Sqrt(1 + Power(k,2)/Power(c - Complex(0,1)*k0,2)))) + 10*(1/k - 1/(k*Sqrt(1 + Power(k,2)/Power(2*c - Complex(0,1)*k0,2)))) - 10*(1/k - 1/(k*Sqrt(1 + Power(k,2)/Power(3*c - Complex(0,1)*k0,2)))) + 5*(1/k - 1/(k*Sqrt(1 + Power(k,2)/Power(4*c - Complex(0,1)*k0,2)))) + 1/(k*Sqrt(1 + Power(k,2)/Power(5*c - Complex(0,1)*k0,2))) + (1 + (Complex(0,1)*k0)/Sqrt(Power(k,2) - Power(k0,2)))/k)/k0
SeriesData[k, Infinity, {-I/k, I, (-I/2*k0^2)/k, I/2*k0^2, (3*(120*c^5 - I*k0^5))/(8*k*k0), (3*I)/8*k0^4, (-5*(16800*c^7 - (12600*I)*c^6*k0 - 2520*c^5*k0^2 + I*k0^7))/(16*k*k0), (5*I)/16*k0^6, (35*(834120*c^9 - (1134000*I)*c^8*k0 - 604800*c^7*k0^2 + (151200*I)*c^6*k0^3 + 15120*c^5*k0^4 - I*k0^9))/(128*k*k0), (35*I)/128*k0^8}, 1, 11, 1]
(-5*(1 - 1/Sqrt[1 + k^2/(c - I*k0)^2]) + 10*(1 - 1/Sqrt[1 + k^2/(2*c - I*k0)^2]) - 10*(1 - 1/Sqrt[1 + k^2/(3*c - I*k0)^2]) + 5*(1 - 1/Sqrt[1 + k^2/(4*c - I*k0)^2]) + 1/Sqrt[1 + k^2/(5*c - I*k0)^2] + (I*k0)/Sqrt[k^2 - k0^2])/(k*k0)

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(-5*(1/(Sqrt[1 + k^2/(c - I*k0)^2]*(c - I*k0)) - (2*(c - I*k0))/k^2 + (2*(c - I*k0))/(k^2*Sqrt[1 + k^2/(c - I*k0)^2])) + 10*(1/(Sqrt[1 + k^2/(2*c - I*k0)^2]*(2*c - I*k0)) - (2*(2*c - I*k0))/k^2 + (2*(2*c - I*k0))/(k^2*Sqrt[1 + k^2/(2*c - I*k0)^2])) - 10*(1/(Sqrt[1 + k^2/(3*c - I*k0)^2]*(3*c - I*k0)) - (2*(3*c - I*k0))/k^2 + (2*(3*c - I*k0))/(k^2*Sqrt[1 + k^2/(3*c - I*k0)^2])) + 5*(1/(Sqrt[1 + k^2/(4*c - I*k0)^2]*(4*c - I*k0)) - (2*(4*c - I*k0))/k^2 + (2*(4*c - I*k0))/(k^2*Sqrt[1 + k^2/(4*c - I*k0)^2])) - 1/(Sqrt[1 + k^2/(5*c - I*k0)^2]*(5*c - I*k0)) + (2*(5*c - I*k0))/k^2 - (2*(5*c - I*k0))/(k^2*Sqrt[1 + k^2/(5*c - I*k0)^2]) + ((2*I)*k*k0 + (k*(k^2 - 2*k0^2))/Sqrt[k^2 - k0^2])/k^3)/k0
(-5*(1/(Sqrt(1 + Power(k,2)/Power(c - Complex(0,1)*k0,2))*(c - Complex(0,1)*k0)) - (2*(c - Complex(0,1)*k0))/Power(k,2) + (2*(c - Complex(0,1)*k0))/(Power(k,2)*Sqrt(1 + Power(k,2)/Power(c - Complex(0,1)*k0,2)))) + 10*(1/(Sqrt(1 + Power(k,2)/Power(2*c - Complex(0,1)*k0,2))*(2*c - Complex(0,1)*k0)) - (2*(2*c - Complex(0,1)*k0))/Power(k,2) + (2*(2*c - Complex(0,1)*k0))/(Power(k,2)*Sqrt(1 + Power(k,2)/Power(2*c - Complex(0,1)*k0,2)))) - 10*(1/(Sqrt(1 + Power(k,2)/Power(3*c - Complex(0,1)*k0,2))*(3*c - Complex(0,1)*k0)) - (2*(3*c - Complex(0,1)*k0))/Power(k,2) + (2*(3*c - Complex(0,1)*k0))/(Power(k,2)*Sqrt(1 + Power(k,2)/Power(3*c - Complex(0,1)*k0,2)))) + 5*(1/(Sqrt(1 + Power(k,2)/Power(4*c - Complex(0,1)*k0,2))*(4*c - Complex(0,1)*k0)) - (2*(4*c - Complex(0,1)*k0))/Power(k,2) + (2*(4*c - Complex(0,1)*k0))/(Power(k,2)*Sqrt(1 + Power(k,2)/Power(4*c - Complex(0,1)*k0,2)))) - 1/(Sqrt(1 + Power(k,2)/Power(5*c - Complex(0,1)*k0,2))*(5*c - Complex(0,1)*k0)) + (2*(5*c - Complex(0,1)*k0))/Power(k,2) - (2*(5*c - Complex(0,1)*k0))/(Power(k,2)*Sqrt(1 + Power(k,2)/Power(5*c - Complex(0,1)*k0,2))) + (Complex(0,2)*k*k0 + (k*(Power(k,2) - 2*Power(k0,2)))/Sqrt(Power(k,2) - Power(k0,2)))/Power(k,3))/k0
SeriesData[k, Infinity, {-(1/(k*k0)), k0^(-1), (3*k0)/(2*k), (-3*k0)/2, ((5*c - I*k0)^4/k - (3*(5*c - I*k0)^3*((5*c)/k - (I*k0)/k))/8 - 5*(-((c - I*k0)^4/k) + (3*(c - I*k0)^3*(c/k - (I*k0)/k))/8) + 10*(-((2*c - I*k0)^4/k) + (3*(2*c - I*k0)^3*((2*c)/k - (I*k0)/k))/8) - 10*(-((3*c - I*k0)^4/k) + (3*(3*c - I*k0)^3*((3*c)/k - (I*k0)/k))/8) + 5*(-((4*c - I*k0)^4/k) + (3*(4*c - I*k0)^3*((4*c)/k - (I*k0)/k))/8))/k0, (-5*k0^3)/8, ((-3*(5*c - I*k0)^6)/(4*k) + (5*(5*c - I*k0)^5*((5*c)/k - (I*k0)/k))/16 - 5*((3*(c - I*k0)^6)/(4*k) - (5*(c - I*k0)^5*(c/k - (I*k0)/k))/16) + 10*((3*(2*c - I*k0)^6)/(4*k) - (5*(2*c - I*k0)^5*((2*c)/k - (I*k0)/k))/16) - 10*((3*(3*c - I*k0)^6)/(4*k) - (5*(3*c - I*k0)^5*((3*c)/k - (I*k0)/k))/16) + 5*((3*(4*c - I*k0)^6)/(4*k) - (5*(4*c - I*k0)^5*((4*c)/k - (I*k0)/k))/16))/k0, (-7*k0^5)/16, ((5*(5*c - I*k0)^8)/(8*k) - (35*(5*c - I*k0)^7*((5*c)/k - (I*k0)/k))/128 - 5*((-5*(c - I*k0)^8)/(8*k) + (35*(c - I*k0)^7*(c/k - (I*k0)/k))/128) + 10*((-5*(2*c - I*k0)^8)/(8*k) + (35*(2*c - I*k0)^7*((2*c)/k - (I*k0)/k))/128) - 10*((-5*(3*c - I*k0)^8)/(8*k) + (35*(3*c - I*k0)^7*((3*c)/k - (I*k0)/k))/128) + 5*((-5*(4*c - I*k0)^8)/(8*k) + (35*(4*c - I*k0)^7*((4*c)/k - (I*k0)/k))/128))/k0, (-45*k0^7)/128, ((-35*(5*c - I*k0)^10)/(64*k) + (63*(5*c - I*k0)^9*((5*c)/k - (I*k0)/k))/256 - 5*((35*(c - I*k0)^10)/(64*k) - (63*(c - I*k0)^9*(c/k - (I*k0)/k))/256) + 10*((35*(2*c - I*k0)^10)/(64*k) - (63*(2*c - I*k0)^9*((2*c)/k - (I*k0)/k))/256) - 10*((35*(3*c - I*k0)^10)/(64*k) - (63*(3*c - I*k0)^9*((3*c)/k - (I*k0)/k))/256) + 5*((35*(4*c - I*k0)^10)/(64*k) - (63*(4*c - I*k0)^9*((4*c)/k - (I*k0)/k))/256))/k0}, 0, 11, 1]
(-5*(1/(Sqrt[1 + k^2/(c - I*k0)^2]*(c - I*k0)) - (2*(c - I*k0))/k^2 + (2*(c - I*k0))/(k^2*Sqrt[1 + k^2/(c - I*k0)^2])) + 5*(1/(Sqrt[1 + k^2/(4*c - I*k0)^2]*(4*c - I*k0)) + (8*c - (2*I)*k0)/(k^2*Sqrt[1 + k^2/(4*c - I*k0)^2]) + (-8*c + (2*I)*k0)/k^2) - 10*(1/(Sqrt[1 + k^2/(3*c - I*k0)^2]*(3*c - I*k0)) + (6*c - (2*I)*k0)/(k^2*Sqrt[1 + k^2/(3*c - I*k0)^2]) + (-6*c + (2*I)*k0)/k^2) + 10*(1/(Sqrt[1 + k^2/(2*c - I*k0)^2]*(2*c - I*k0)) + (4*c - (2*I)*k0)/(k^2*Sqrt[1 + k^2/(2*c - I*k0)^2]) + (-4*c + (2*I)*k0)/k^2) - 1/(Sqrt[1 + k^2/(5*c - I*k0)^2]*(5*c - I*k0)) + (10*c - (2*I)*k0)/k^2 + (-10*c + (2*I)*k0)/(k^2*Sqrt[1 + k^2/(5*c - I*k0)^2]) + ((2*I)*k*k0 + (k*(k^2 - 2*k0^2))/Sqrt[k^2 - k0^2])/k^3)/k0

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(-5*(-1 + Sqrt[1 + k^2/(c - I*k0)^2])*(c - I*k0) + 10*(-1 + Sqrt[1 + k^2/(2*c - I*k0)^2])*(2*c - I*k0) - 10*(-1 + Sqrt[1 + k^2/(3*c - I*k0)^2])*(3*c - I*k0) + 5*(-1 + Sqrt[1 + k^2/(4*c - I*k0)^2])*(4*c - I*k0) - (-1 + Sqrt[1 + k^2/(5*c - I*k0)^2])*(5*c - I*k0) + I*k0 + Sqrt[k^2 - k0^2])/(k*k0^2)
(-5*(-1 + Sqrt(1 + Power(k,2)/Power(c - Complex(0,1)*k0,2)))*(c - Complex(0,1)*k0) + 10*(-1 + Sqrt(1 + Power(k,2)/Power(2*c - Complex(0,1)*k0,2)))*(2*c - Complex(0,1)*k0) - 10*(-1 + Sqrt(1 + Power(k,2)/Power(3*c - Complex(0,1)*k0,2)))*(3*c - Complex(0,1)*k0) + 5*(-1 + Sqrt(1 + Power(k,2)/Power(4*c - Complex(0,1)*k0,2)))*(4*c - Complex(0,1)*k0) - (-1 + Sqrt(1 + Power(k,2)/Power(5*c - Complex(0,1)*k0,2)))*(5*c - Complex(0,1)*k0) + Complex(0,1)*k0 + Sqrt(Power(k,2) - Power(k0,2)))/(k*Power(k0,2))
SeriesData[k, Infinity, {(-225*c^6)/(2*k0^2) + ((45*I)*c^5)/k0, 0, (-7875*c^6)/4 + (39375*c^8)/(8*k0^2) - ((5250*I)*c^7)/k0 + (525*I)/2*c^5*k0, 0, (-2205*(2025*c^10 - (3310*I)*c^9*k0 - 2250*c^8*k0^2 + (800*I)*c^7*k0^3 + 150*c^6*k0^4 - (12*I)*c^5*k0^5))/(32*k0^2)}, 6, 11, 1]
(-5*(-1 + Sqrt[1 + k^2/(c - I*k0)^2])*(c - I*k0) + 10*(-1 + Sqrt[1 + k^2/(2*c - I*k0)^2])*(2*c - I*k0) - 10*(-1 + Sqrt[1 + k^2/(3*c - I*k0)^2])*(3*c - I*k0) + 5*(-1 + Sqrt[1 + k^2/(4*c - I*k0)^2])*(4*c - I*k0) - (-1 + Sqrt[1 + k^2/(5*c - I*k0)^2])*(5*c - I*k0) + I*k0 + Sqrt[k^2 - k0^2])/(k*k0^2)

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(5*(-1 + Sqrt[1 + k^2/(c - I*k0)^2])*(c - I*k0)^2 - 10*(-1 + Sqrt[1 + k^2/(2*c - I*k0)^2])*(2*c - I*k0)^2 + 10*(-1 + Sqrt[1 + k^2/(3*c - I*k0)^2])*(3*c - I*k0)^2 - 5*(-1 + Sqrt[1 + k^2/(4*c - I*k0)^2])*(4*c - I*k0)^2 + (-1 + Sqrt[1 + k^2/(5*c - I*k0)^2])*(5*c - I*k0)^2 - k0^2 + I*k0*Sqrt[k^2 - k0^2])/(k^2*k0^2)
(5*(-1 + Sqrt(1 + Power(k,2)/Power(c - Complex(0,1)*k0,2)))*Power(c - Complex(0,1)*k0,2) - 10*(-1 + Sqrt(1 + Power(k,2)/Power(2*c - Complex(0,1)*k0,2)))*Power(2*c - Complex(0,1)*k0,2) + 10*(-1 + Sqrt(1 + Power(k,2)/Power(3*c - Complex(0,1)*k0,2)))*Power(3*c - Complex(0,1)*k0,2) - 5*(-1 + Sqrt(1 + Power(k,2)/Power(4*c - Complex(0,1)*k0,2)))*Power(4*c - Complex(0,1)*k0,2) + (-1 + Sqrt(1 + Power(k,2)/Power(5*c - Complex(0,1)*k0,2)))*Power(5*c - Complex(0,1)*k0,2) - Power(k0,2) + Complex(0,1)*k0*Sqrt(Power(k,2) - Power(k0,2)))/(Power(k,2)*Power(k0,2))
SeriesData[k, Infinity, {(-15*c^5)/k0^2, 0, (-315*c^5)/2 + (1050*c^7)/k0^2 - ((1575*I)/2*c^6)/k0, 0, (-1575*(331*c^9 - (450*I)*c^8*k0 - 240*c^7*k0^2 + (60*I)*c^6*k0^3 + 6*c^5*k0^4))/(16*k0^2)}, 5, 11, 1]
(5*(-1 + Sqrt[1 + k^2/(c - I*k0)^2])*(c - I*k0)^2 - 10*(-1 + Sqrt[1 + k^2/(2*c - I*k0)^2])*(2*c - I*k0)^2 + 10*(-1 + Sqrt[1 + k^2/(3*c - I*k0)^2])*(3*c - I*k0)^2 - 5*(-1 + Sqrt[1 + k^2/(4*c - I*k0)^2])*(4*c - I*k0)^2 + (-1 + Sqrt[1 + k^2/(5*c - I*k0)^2])*(5*c - I*k0)^2 - k0^2 + I*k0*Sqrt[k^2 - k0^2])/(k^2*k0^2)

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besseltransforms/klarge/runit.sh Executable file
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#!/bin/bash
K=$1
Q=$2
N=$3
module load mathematica
cat - vzor.m <<<"
kk=$K;
qq=$Q;
nn=$N;
" | math -noprompt > "${K}-${Q}-${N}"

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besseltransforms/klarge/vzor.m Normal file
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$Assumptions = k >= 0 && k > k0 && k0 >= 0 && c >= 0 && n >= 0 ;
f = Refine[Integrate[(1 - Exp[-c x])^\[Kappa] (k0 x)^(-q) Exp[
I k0 x] x BesselJ[n, k x], {x,
0, \[Infinity]}], {\[Kappa] == kk, q == qq, n == nn}]
CForm[f]
Series[f, {k, \[Infinity], 10}]
Simplify[f]
Quit[ ]

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(-5/(Sqrt[1 + k^2/(c - I*k0)^2]*(c - I*k0)) + 10/(Sqrt[1 + k^2/(2*c - I*k0)^2]*(2*c - I*k0)) - 10/(Sqrt[1 + k^2/(3*c - I*k0)^2]*(3*c - I*k0)) + 5/(Sqrt[1 + k^2/(4*c - I*k0)^2]*(4*c - I*k0)) - 1/(Sqrt[1 + k^2/(5*c - I*k0)^2]*(5*c - I*k0)) + I/Sqrt[-k^2 + k0^2])/k0
(-5/(Sqrt(1 + Power(k,2)/Power(c - Complex(0,1)*k0,2))*(c - Complex(0,1)*k0)) + 10/(Sqrt(1 + Power(k,2)/Power(2*c - Complex(0,1)*k0,2))*(2*c - Complex(0,1)*k0)) - 10/(Sqrt(1 + Power(k,2)/Power(3*c - Complex(0,1)*k0,2))*(3*c - Complex(0,1)*k0)) + 5/(Sqrt(1 + Power(k,2)/Power(4*c - Complex(0,1)*k0,2))*(4*c - Complex(0,1)*k0)) - 1/(Sqrt(1 + Power(k,2)/Power(5*c - Complex(0,1)*k0,2))*(5*c - Complex(0,1)*k0)) + Complex(0,1)/Sqrt(-Power(k,2) + Power(k0,2)))/k0
SeriesData[k, Infinity, {-(1/(k*k0)), k0^(-1), -k0/(2*k), k0/2, ((-15*(c - I*k0)^3*(c/k - (I*k0)/k))/8 + (15*(2*c - I*k0)^3*((2*c)/k - (I*k0)/k))/4 - (15*(3*c - I*k0)^3*((3*c)/k - (I*k0)/k))/4 + (15*(4*c - I*k0)^3*((4*c)/k - (I*k0)/k))/8 - (3*(5*c - I*k0)^3*((5*c)/k - (I*k0)/k))/8)/k0, (3*k0^3)/8, ((25*(c - I*k0)^5*(c/k - (I*k0)/k))/16 - (25*(2*c - I*k0)^5*((2*c)/k - (I*k0)/k))/8 + (25*(3*c - I*k0)^5*((3*c)/k - (I*k0)/k))/8 - (25*(4*c - I*k0)^5*((4*c)/k - (I*k0)/k))/16 + (5*(5*c - I*k0)^5*((5*c)/k - (I*k0)/k))/16)/k0, (5*k0^5)/16, ((-175*(c - I*k0)^7*(c/k - (I*k0)/k))/128 + (175*(2*c - I*k0)^7*((2*c)/k - (I*k0)/k))/64 - (175*(3*c - I*k0)^7*((3*c)/k - (I*k0)/k))/64 + (175*(4*c - I*k0)^7*((4*c)/k - (I*k0)/k))/128 - (35*(5*c - I*k0)^7*((5*c)/k - (I*k0)/k))/128)/k0, (35*k0^7)/128, ((315*(c - I*k0)^9*(c/k - (I*k0)/k))/256 - (315*(2*c - I*k0)^9*((2*c)/k - (I*k0)/k))/128 + (315*(3*c - I*k0)^9*((3*c)/k - (I*k0)/k))/128 - (315*(4*c - I*k0)^9*((4*c)/k - (I*k0)/k))/256 + (63*(5*c - I*k0)^9*((5*c)/k - (I*k0)/k))/256)/k0}, 0, 11, 1]
(-5/(Sqrt[1 + k^2/(c - I*k0)^2]*(c - I*k0)) + 10/(Sqrt[1 + k^2/(2*c - I*k0)^2]*(2*c - I*k0)) - 10/(Sqrt[1 + k^2/(3*c - I*k0)^2]*(3*c - I*k0)) + 5/(Sqrt[1 + k^2/(4*c - I*k0)^2]*(4*c - I*k0)) - 1/(Sqrt[1 + k^2/(5*c - I*k0)^2]*(5*c - I*k0)) + I/Sqrt[-k^2 + k0^2])/k0

4
besseltransforms/ksmall/5-1-1 Normal file
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(-k^(-1) - 5*(k^(-1) - 1/(k*Sqrt[1 + k^2/(c - I*k0)^2])) + 10*(k^(-1) - 1/(k*Sqrt[1 + k^2/(2*c - I*k0)^2])) - 10*(k^(-1) - 1/(k*Sqrt[1 + k^2/(3*c - I*k0)^2])) + 5*(k^(-1) - 1/(k*Sqrt[1 + k^2/(4*c - I*k0)^2])) + 1/(k*Sqrt[1 + k^2/(5*c - I*k0)^2]) + (1 - k0/Sqrt[-k^2 + k0^2])/k)/k0
(-(1/k) - 5*(1/k - 1/(k*Sqrt(1 + Power(k,2)/Power(c - Complex(0,1)*k0,2)))) + 10*(1/k - 1/(k*Sqrt(1 + Power(k,2)/Power(2*c - Complex(0,1)*k0,2)))) - 10*(1/k - 1/(k*Sqrt(1 + Power(k,2)/Power(3*c - Complex(0,1)*k0,2)))) + 5*(1/k - 1/(k*Sqrt(1 + Power(k,2)/Power(4*c - Complex(0,1)*k0,2)))) + 1/(k*Sqrt(1 + Power(k,2)/Power(5*c - Complex(0,1)*k0,2))) + (1 - k0/Sqrt(-Power(k,2) + Power(k0,2)))/k)/k0
SeriesData[k, Infinity, {-I/k, I, (-I/2*k0^2)/k, I/2*k0^2, (3*(120*c^5 - I*k0^5))/(8*k*k0), (3*I)/8*k0^4, (-5*(16800*c^7 - (12600*I)*c^6*k0 - 2520*c^5*k0^2 + I*k0^7))/(16*k*k0), (5*I)/16*k0^6, (35*(834120*c^9 - (1134000*I)*c^8*k0 - 604800*c^7*k0^2 + (151200*I)*c^6*k0^3 + 15120*c^5*k0^4 - I*k0^9))/(128*k*k0), (35*I)/128*k0^8}, 1, 11, 1]
(-5*(1 - 1/Sqrt[1 + k^2/(c - I*k0)^2]) + 10*(1 - 1/Sqrt[1 + k^2/(2*c - I*k0)^2]) - 10*(1 - 1/Sqrt[1 + k^2/(3*c - I*k0)^2]) + 5*(1 - 1/Sqrt[1 + k^2/(4*c - I*k0)^2]) + 1/Sqrt[1 + k^2/(5*c - I*k0)^2] - k0/Sqrt[-k^2 + k0^2])/(k*k0)

4
besseltransforms/ksmall/5-1-2 Normal file
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(-5*(1/(Sqrt[1 + k^2/(c - I*k0)^2]*(c - I*k0)) - (2*(c - I*k0))/k^2 + (2*(c - I*k0))/(k^2*Sqrt[1 + k^2/(c - I*k0)^2])) + 10*(1/(Sqrt[1 + k^2/(2*c - I*k0)^2]*(2*c - I*k0)) - (2*(2*c - I*k0))/k^2 + (2*(2*c - I*k0))/(k^2*Sqrt[1 + k^2/(2*c - I*k0)^2])) - 10*(1/(Sqrt[1 + k^2/(3*c - I*k0)^2]*(3*c - I*k0)) - (2*(3*c - I*k0))/k^2 + (2*(3*c - I*k0))/(k^2*Sqrt[1 + k^2/(3*c - I*k0)^2])) + 5*(1/(Sqrt[1 + k^2/(4*c - I*k0)^2]*(4*c - I*k0)) - (2*(4*c - I*k0))/k^2 + (2*(4*c - I*k0))/(k^2*Sqrt[1 + k^2/(4*c - I*k0)^2])) - 1/(Sqrt[1 + k^2/(5*c - I*k0)^2]*(5*c - I*k0)) + (2*(5*c - I*k0))/k^2 - (2*(5*c - I*k0))/(k^2*Sqrt[1 + k^2/(5*c - I*k0)^2]) + (I*(k^2 + 2*k0*(-k0 + Sqrt[-k^2 + k0^2])))/(k^2*Sqrt[-k^2 + k0^2]))/k0
(-5*(1/(Sqrt(1 + Power(k,2)/Power(c - Complex(0,1)*k0,2))*(c - Complex(0,1)*k0)) - (2*(c - Complex(0,1)*k0))/Power(k,2) + (2*(c - Complex(0,1)*k0))/(Power(k,2)*Sqrt(1 + Power(k,2)/Power(c - Complex(0,1)*k0,2)))) + 10*(1/(Sqrt(1 + Power(k,2)/Power(2*c - Complex(0,1)*k0,2))*(2*c - Complex(0,1)*k0)) - (2*(2*c - Complex(0,1)*k0))/Power(k,2) + (2*(2*c - Complex(0,1)*k0))/(Power(k,2)*Sqrt(1 + Power(k,2)/Power(2*c - Complex(0,1)*k0,2)))) - 10*(1/(Sqrt(1 + Power(k,2)/Power(3*c - Complex(0,1)*k0,2))*(3*c - Complex(0,1)*k0)) - (2*(3*c - Complex(0,1)*k0))/Power(k,2) + (2*(3*c - Complex(0,1)*k0))/(Power(k,2)*Sqrt(1 + Power(k,2)/Power(3*c - Complex(0,1)*k0,2)))) + 5*(1/(Sqrt(1 + Power(k,2)/Power(4*c - Complex(0,1)*k0,2))*(4*c - Complex(0,1)*k0)) - (2*(4*c - Complex(0,1)*k0))/Power(k,2) + (2*(4*c - Complex(0,1)*k0))/(Power(k,2)*Sqrt(1 + Power(k,2)/Power(4*c - Complex(0,1)*k0,2)))) - 1/(Sqrt(1 + Power(k,2)/Power(5*c - Complex(0,1)*k0,2))*(5*c - Complex(0,1)*k0)) + (2*(5*c - Complex(0,1)*k0))/Power(k,2) - (2*(5*c - Complex(0,1)*k0))/(Power(k,2)*Sqrt(1 + Power(k,2)/Power(5*c - Complex(0,1)*k0,2))) + (Complex(0,1)*(Power(k,2) + 2*k0*(-k0 + Sqrt(-Power(k,2) + Power(k0,2)))))/(Power(k,2)*Sqrt(-Power(k,2) + Power(k0,2))))/k0
SeriesData[k, Infinity, {-(1/(k*k0)), k0^(-1), (3*k0)/(2*k), (-3*k0)/2, ((5*c - I*k0)^4/k - (3*(5*c - I*k0)^3*((5*c)/k - (I*k0)/k))/8 - 5*(-((c - I*k0)^4/k) + (3*(c - I*k0)^3*(c/k - (I*k0)/k))/8) + 10*(-((2*c - I*k0)^4/k) + (3*(2*c - I*k0)^3*((2*c)/k - (I*k0)/k))/8) - 10*(-((3*c - I*k0)^4/k) + (3*(3*c - I*k0)^3*((3*c)/k - (I*k0)/k))/8) + 5*(-((4*c - I*k0)^4/k) + (3*(4*c - I*k0)^3*((4*c)/k - (I*k0)/k))/8))/k0, (-5*k0^3)/8, ((-3*(5*c - I*k0)^6)/(4*k) + (5*(5*c - I*k0)^5*((5*c)/k - (I*k0)/k))/16 - 5*((3*(c - I*k0)^6)/(4*k) - (5*(c - I*k0)^5*(c/k - (I*k0)/k))/16) + 10*((3*(2*c - I*k0)^6)/(4*k) - (5*(2*c - I*k0)^5*((2*c)/k - (I*k0)/k))/16) - 10*((3*(3*c - I*k0)^6)/(4*k) - (5*(3*c - I*k0)^5*((3*c)/k - (I*k0)/k))/16) + 5*((3*(4*c - I*k0)^6)/(4*k) - (5*(4*c - I*k0)^5*((4*c)/k - (I*k0)/k))/16))/k0, (-7*k0^5)/16, ((5*(5*c - I*k0)^8)/(8*k) - (35*(5*c - I*k0)^7*((5*c)/k - (I*k0)/k))/128 - 5*((-5*(c - I*k0)^8)/(8*k) + (35*(c - I*k0)^7*(c/k - (I*k0)/k))/128) + 10*((-5*(2*c - I*k0)^8)/(8*k) + (35*(2*c - I*k0)^7*((2*c)/k - (I*k0)/k))/128) - 10*((-5*(3*c - I*k0)^8)/(8*k) + (35*(3*c - I*k0)^7*((3*c)/k - (I*k0)/k))/128) + 5*((-5*(4*c - I*k0)^8)/(8*k) + (35*(4*c - I*k0)^7*((4*c)/k - (I*k0)/k))/128))/k0, (-45*k0^7)/128, ((-35*(5*c - I*k0)^10)/(64*k) + (63*(5*c - I*k0)^9*((5*c)/k - (I*k0)/k))/256 - 5*((35*(c - I*k0)^10)/(64*k) - (63*(c - I*k0)^9*(c/k - (I*k0)/k))/256) + 10*((35*(2*c - I*k0)^10)/(64*k) - (63*(2*c - I*k0)^9*((2*c)/k - (I*k0)/k))/256) - 10*((35*(3*c - I*k0)^10)/(64*k) - (63*(3*c - I*k0)^9*((3*c)/k - (I*k0)/k))/256) + 5*((35*(4*c - I*k0)^10)/(64*k) - (63*(4*c - I*k0)^9*((4*c)/k - (I*k0)/k))/256))/k0}, 0, 11, 1]
(-5*(1/(Sqrt[1 + k^2/(c - I*k0)^2]*(c - I*k0)) - (2*(c - I*k0))/k^2 + (2*(c - I*k0))/(k^2*Sqrt[1 + k^2/(c - I*k0)^2])) + 5*(1/(Sqrt[1 + k^2/(4*c - I*k0)^2]*(4*c - I*k0)) + (8*c - (2*I)*k0)/(k^2*Sqrt[1 + k^2/(4*c - I*k0)^2]) + (-8*c + (2*I)*k0)/k^2) - 10*(1/(Sqrt[1 + k^2/(3*c - I*k0)^2]*(3*c - I*k0)) + (6*c - (2*I)*k0)/(k^2*Sqrt[1 + k^2/(3*c - I*k0)^2]) + (-6*c + (2*I)*k0)/k^2) + 10*(1/(Sqrt[1 + k^2/(2*c - I*k0)^2]*(2*c - I*k0)) + (4*c - (2*I)*k0)/(k^2*Sqrt[1 + k^2/(2*c - I*k0)^2]) + (-4*c + (2*I)*k0)/k^2) - 1/(Sqrt[1 + k^2/(5*c - I*k0)^2]*(5*c - I*k0)) + (10*c - (2*I)*k0)/k^2 + (-10*c + (2*I)*k0)/(k^2*Sqrt[1 + k^2/(5*c - I*k0)^2]) + (I*(k^2 + 2*k0*(-k0 + Sqrt[-k^2 + k0^2])))/(k^2*Sqrt[-k^2 + k0^2]))/k0

0
besseltransforms/ksmall/5-2-0 Normal file
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4
besseltransforms/ksmall/5-2-1 Normal file
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(-5*(-1 + Sqrt[1 + k^2/(c - I*k0)^2])*(c - I*k0) + 10*(-1 + Sqrt[1 + k^2/(2*c - I*k0)^2])*(2*c - I*k0) - 10*(-1 + Sqrt[1 + k^2/(3*c - I*k0)^2])*(3*c - I*k0) + 5*(-1 + Sqrt[1 + k^2/(4*c - I*k0)^2])*(4*c - I*k0) - (-1 + Sqrt[1 + k^2/(5*c - I*k0)^2])*(5*c - I*k0) + I*(k0 - Sqrt[-k^2 + k0^2]))/(k*k0^2)
(-5*(-1 + Sqrt(1 + Power(k,2)/Power(c - Complex(0,1)*k0,2)))*(c - Complex(0,1)*k0) + 10*(-1 + Sqrt(1 + Power(k,2)/Power(2*c - Complex(0,1)*k0,2)))*(2*c - Complex(0,1)*k0) - 10*(-1 + Sqrt(1 + Power(k,2)/Power(3*c - Complex(0,1)*k0,2)))*(3*c - Complex(0,1)*k0) + 5*(-1 + Sqrt(1 + Power(k,2)/Power(4*c - Complex(0,1)*k0,2)))*(4*c - Complex(0,1)*k0) - (-1 + Sqrt(1 + Power(k,2)/Power(5*c - Complex(0,1)*k0,2)))*(5*c - Complex(0,1)*k0) + Complex(0,1)*(k0 - Sqrt(-Power(k,2) + Power(k0,2))))/(k*Power(k0,2))
SeriesData[k, Infinity, {(-225*c^6)/(2*k0^2) + ((45*I)*c^5)/k0, 0, (-7875*c^6)/4 + (39375*c^8)/(8*k0^2) - ((5250*I)*c^7)/k0 + (525*I)/2*c^5*k0, 0, (-2205*(2025*c^10 - (3310*I)*c^9*k0 - 2250*c^8*k0^2 + (800*I)*c^7*k0^3 + 150*c^6*k0^4 - (12*I)*c^5*k0^5))/(32*k0^2)}, 6, 11, 1]
(-5*(-1 + Sqrt[1 + k^2/(c - I*k0)^2])*(c - I*k0) + 10*(-1 + Sqrt[1 + k^2/(2*c - I*k0)^2])*(2*c - I*k0) - 10*(-1 + Sqrt[1 + k^2/(3*c - I*k0)^2])*(3*c - I*k0) + 5*(-1 + Sqrt[1 + k^2/(4*c - I*k0)^2])*(4*c - I*k0) - (-1 + Sqrt[1 + k^2/(5*c - I*k0)^2])*(5*c - I*k0) + I*(k0 - Sqrt[-k^2 + k0^2]))/(k*k0^2)

4
besseltransforms/ksmall/5-2-2 Normal file
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(5*(-1 + Sqrt[1 + k^2/(c - I*k0)^2])*(c - I*k0)^2 - 10*(-1 + Sqrt[1 + k^2/(2*c - I*k0)^2])*(2*c - I*k0)^2 + 10*(-1 + Sqrt[1 + k^2/(3*c - I*k0)^2])*(3*c - I*k0)^2 - 5*(-1 + Sqrt[1 + k^2/(4*c - I*k0)^2])*(4*c - I*k0)^2 + (-1 + Sqrt[1 + k^2/(5*c - I*k0)^2])*(5*c - I*k0)^2 - k0*(k0 - Sqrt[-k^2 + k0^2]))/(k^2*k0^2)
(5*(-1 + Sqrt(1 + Power(k,2)/Power(c - Complex(0,1)*k0,2)))*Power(c - Complex(0,1)*k0,2) - 10*(-1 + Sqrt(1 + Power(k,2)/Power(2*c - Complex(0,1)*k0,2)))*Power(2*c - Complex(0,1)*k0,2) + 10*(-1 + Sqrt(1 + Power(k,2)/Power(3*c - Complex(0,1)*k0,2)))*Power(3*c - Complex(0,1)*k0,2) - 5*(-1 + Sqrt(1 + Power(k,2)/Power(4*c - Complex(0,1)*k0,2)))*Power(4*c - Complex(0,1)*k0,2) + (-1 + Sqrt(1 + Power(k,2)/Power(5*c - Complex(0,1)*k0,2)))*Power(5*c - Complex(0,1)*k0,2) - k0*(k0 - Sqrt(-Power(k,2) + Power(k0,2))))/(Power(k,2)*Power(k0,2))
SeriesData[k, Infinity, {(-15*c^5)/k0^2, 0, (-315*c^5)/2 + (1050*c^7)/k0^2 - ((1575*I)/2*c^6)/k0, 0, (-1575*(331*c^9 - (450*I)*c^8*k0 - 240*c^7*k0^2 + (60*I)*c^6*k0^3 + 6*c^5*k0^4))/(16*k0^2)}, 5, 11, 1]
(5*(-1 + Sqrt[1 + k^2/(c - I*k0)^2])*(c - I*k0)^2 - 10*(-1 + Sqrt[1 + k^2/(2*c - I*k0)^2])*(2*c - I*k0)^2 + 10*(-1 + Sqrt[1 + k^2/(3*c - I*k0)^2])*(3*c - I*k0)^2 - 5*(-1 + Sqrt[1 + k^2/(4*c - I*k0)^2])*(4*c - I*k0)^2 + (-1 + Sqrt[1 + k^2/(5*c - I*k0)^2])*(5*c - I*k0)^2 - k0*(k0 - Sqrt[-k^2 + k0^2]))/(k^2*k0^2)

11
besseltransforms/ksmall/runit.sh Executable file
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@ -0,0 +1,11 @@
#!/bin/bash
K=$1
Q=$2
N=$3
module load mathematica
cat - vzor.m <<<"
kk=$K;
qq=$Q;
nn=$N;
" | math -noprompt > "${K}-${Q}-${N}"

8
besseltransforms/ksmall/vzor.m Normal file
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@ -0,0 +1,8 @@
$Assumptions = k >= 0 && k < k0 && k0 >= 0 && c >= 0 && n >= 0 ;
f = Refine[Integrate[(1 - Exp[-c x])^\[Kappa] (k0 x)^(-q) Exp[
I k0 x] x BesselJ[n, k x], {x,
0, \[Infinity]}], {\[Kappa] == kk, q == qq, n == nn}]
CForm[f]
Series[f, {k, \[Infinity], 10}]
Simplify[f]
Quit[ ]

4
besseltransforms/vzor.m
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@ -1,6 +1,6 @@
$Assumptions = k >= 0 && k0 >= 0 && c >= 0 && n >= 0 ;
Refine[Integrate[(1 - Exp[-c x])^\[Kappa] (k0 x)^(-q) Exp[
Simplify[Refine[Integrate[(1 - Exp[-c x])^\[Kappa] (k0 x)^(-q) Exp[
I k0 x] x BesselJ[n, k x], {x,
0, \[Infinity]}], {\[Kappa] == kk, q == qq, n == nn}]
0, \[Infinity]}], {\[Kappa] == kk, q == qq, n == nn}]]
Series[%, {k, \[Infinity], 10}]
Quit[ ]