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  "body": "\n## Definition\n\nFix complex parameters $\\kappa$ and $\\mu$, and let $W_{\\kappa, \\mu}$ be the [[Whittaker function|7EFDMJSVDP]] on $(0, \\infty)$.\nLet $k(z) = e^{-z/2} z^{-\\kappa} W_{\\kappa, \\mu}(z)$ for $z \u003e 0$.\nLet $f \\colon (0, \\infty) \\to \\C$ be such that the integral below converges for $y$ in a set $Y \\subseteq (0, \\infty)$.\nThe *Whittaker transform* of $f$, in this normalization, is the function\n$$g(y) = \\int_0^\\infty k(xy)\\, f(x)\\, dx, \\qquad y \\in Y.$$\n\n## Remarks\n\nOther authors put other powers of $xy$ and other exponential factors in front of $W_{\\kappa, \\mu}(xy)$; the name covers this whole family of kernels.\n\nExample: for $\\kappa = \\mu + 1/2$ one has $W_{\\mu + 1/2, \\mu}(z) = e^{-z/2} z^{\\mu + 1/2}$.\nThen $k(z) = e^{-z/2} z^{-\\mu - 1/2} e^{-z/2} z^{\\mu + 1/2} = e^{-z}$, and the transform is the [[Laplace transform|0V49JEZW06]] $g(y) = \\int_0^\\infty e^{-xy} f(x)\\, dx$.\n\nNon-example: $W_{\\kappa, \\mu}(z)$ behaves like $e^{-z/2} z^{\\kappa}$ as $z \\to \\infty$, so $k(xy)$ behaves like $e^{-xy}$ for large $x$.\nHence $f(x) = e^{x^2}$ gives a divergent integral for every $y \u003e 0$ and has no Whittaker transform.\n",
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