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Lommel Differential Equation

Lommel differential equation (common versions)

z^2 w''(z) + z w'(z) + (z^2 - ν^2) w(z) = z^(μ + 1) for (w(z) = c_1 J_ν(z) + c_2 Y_ν(z) + LommelS1(μ, ν, z) and LommelS1(μ, ν, z) = (_1 F_2(1;1/2 (3 + μ + ν), 1/2 (3 + μ - ν);-z^2/4) z^(μ + 1))/((μ + 1)^2 - ν^2) and (not (1/2 (-3 - μ - ν) element Z and 1/2 (-3 - μ - ν)>=0 and 1/2 (-3 - μ + ν) element Z and 1/2 (-3 - μ + ν)>=0))) z^2 w''(z) + z w'(z) + (z^2 - ν^2) w(z) = z^(μ + 1) for (w(z) = c_1 J_ν(z) + c_2 Y_ν(z) + LommelS2(μ, ν, z) and LommelS2(μ, ν, z) = (_1 F_2(1;1/2 (3 + μ + ν), 1/2 (3 + μ - ν);-z^2/4) z^(μ + 1))/((μ + 1)^2 - ν^2) + (0F1(1 - ν, -z^2/4) (2^(-1 + μ + ν) Γ(ν) Γ(1/2 (1 + μ + ν))))/(z^ν Γ(1/2 (1 - μ + ν))) + (0F1(1 + ν, -z^2/4) (2^(-1 + μ - ν) z^ν Γ(-ν) Γ(1/2 (1 + μ - ν))))/Γ(1/2 (1 - μ - ν)) and (not (1/2 (-3 - μ - ν) element Z and 1/2 (-3 - μ - ν)>=0 and 1/2 (-3 - μ + ν) element Z and 1/2 (-3 - μ + ν)>=0 and ν element Z))) z^2 w''(z) + z w'(z) + (z^2 - ν^2) w(z) = z^(μ + 1) for (w(z) = c_1 J_ν(z) + c_2 Y_ν(z) + LommelS1(μ, ν, z) and LomSmelS1(μ, ν, z) = (_1 F_2(1;1/2 (3 + μ + ν), 1/2 (3 + μ - ν);-z^2/4) z^(μ + 1))/((μ + 1)^2 - ν^2) and (not (1/2 (-3 - μ - ν) element Z and 1/2 (-3 - μ - ν)>=0 and 1/2 (-3 - μ + ν) element Z and 1/2 (-3 - μ + ν)>=0)))

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