How do you solve $int_0^-inftyJ_0(br)e^iasqrtf_0^2-r^2rdr$

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How is solved this integral of Bessel function ?
$$int_0^-inftyJ_0(br)e^iasqrtf_0^2-r^2rdr$$



where $a, b$ are real numbers, $f_0>r$










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  • Looks like a Hankel Transform. Maybe the corresponding 2D Fourier-Transform is easier ... ? Otherwise you may expand the Bessel Function in a power-series and then integrate. Also make sure to check out common intagral-tables, e.g. Gradshteyn Ryzhick. Finally: is $e^i a sqrtf_0^2-r^2 in L^1(]0,infty[)$?
    – denklo
    Sep 7 at 10:06














up vote
0
down vote

favorite












How is solved this integral of Bessel function ?
$$int_0^-inftyJ_0(br)e^iasqrtf_0^2-r^2rdr$$



where $a, b$ are real numbers, $f_0>r$










share|cite|improve this question























  • Looks like a Hankel Transform. Maybe the corresponding 2D Fourier-Transform is easier ... ? Otherwise you may expand the Bessel Function in a power-series and then integrate. Also make sure to check out common intagral-tables, e.g. Gradshteyn Ryzhick. Finally: is $e^i a sqrtf_0^2-r^2 in L^1(]0,infty[)$?
    – denklo
    Sep 7 at 10:06












up vote
0
down vote

favorite









up vote
0
down vote

favorite











How is solved this integral of Bessel function ?
$$int_0^-inftyJ_0(br)e^iasqrtf_0^2-r^2rdr$$



where $a, b$ are real numbers, $f_0>r$










share|cite|improve this question















How is solved this integral of Bessel function ?
$$int_0^-inftyJ_0(br)e^iasqrtf_0^2-r^2rdr$$



where $a, b$ are real numbers, $f_0>r$







integration transformation






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edited Sep 7 at 9:57









Bernard

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112k635104










asked Sep 7 at 9:47









ole

153




153











  • Looks like a Hankel Transform. Maybe the corresponding 2D Fourier-Transform is easier ... ? Otherwise you may expand the Bessel Function in a power-series and then integrate. Also make sure to check out common intagral-tables, e.g. Gradshteyn Ryzhick. Finally: is $e^i a sqrtf_0^2-r^2 in L^1(]0,infty[)$?
    – denklo
    Sep 7 at 10:06
















  • Looks like a Hankel Transform. Maybe the corresponding 2D Fourier-Transform is easier ... ? Otherwise you may expand the Bessel Function in a power-series and then integrate. Also make sure to check out common intagral-tables, e.g. Gradshteyn Ryzhick. Finally: is $e^i a sqrtf_0^2-r^2 in L^1(]0,infty[)$?
    – denklo
    Sep 7 at 10:06















Looks like a Hankel Transform. Maybe the corresponding 2D Fourier-Transform is easier ... ? Otherwise you may expand the Bessel Function in a power-series and then integrate. Also make sure to check out common intagral-tables, e.g. Gradshteyn Ryzhick. Finally: is $e^i a sqrtf_0^2-r^2 in L^1(]0,infty[)$?
– denklo
Sep 7 at 10:06




Looks like a Hankel Transform. Maybe the corresponding 2D Fourier-Transform is easier ... ? Otherwise you may expand the Bessel Function in a power-series and then integrate. Also make sure to check out common intagral-tables, e.g. Gradshteyn Ryzhick. Finally: is $e^i a sqrtf_0^2-r^2 in L^1(]0,infty[)$?
– denklo
Sep 7 at 10:06















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