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  "body": "\n## Definition\n\nLet $\\nu \\in \\C$ with $\\nu + \\frac{3}{2}$ not a non-positive integer, let $\\Gamma$ be the [[gamma function|0H5SYBSED9]], and let $z \\in \\C \\setminus (-\\infty, 0]$ with the principal branch of powers.\nThe *Struve function* of order $\\nu$ is\n$$\\mathbf{H}_\\nu(z) = \\sum_{m=0}^{\\infty} \\frac{(-1)^m}{\\Gamma\\left( m + \\frac{3}{2} \\right) \\Gamma\\left( m + \\nu + \\frac{3}{2} \\right)} \\left( \\frac{z}{2} \\right)^{2m + \\nu + 1}.$$\n\n## Remarks\n\nIt is a particular solution of the inhomogeneous Bessel equation\n$$z^2 w'' + z w' + (z^2 - \\nu^2)\\, w = \\frac{4 \\left( z/2 \\right)^{\\nu + 1}}{\\sqrt{\\pi}\\, \\Gamma\\left( \\nu + \\frac{1}{2} \\right)}.$$\nThe general solution adds to $\\mathbf{H}_\\nu$ any solution of Bessel's equation, such as a combination of the [[Bessel functions|CZCMRDWDYW]] $J_\\nu$ and $Y_\\nu$.\nExample: for $\\nu = \\frac{1}{2}$ and $x \u003e 0$, $\\mathbf{H}_{1/2}(x) = \\sqrt{\\frac{2}{\\pi x}}\\,(1 - \\cos x)$; the first term of the series is $\\frac{(x/2)^{3/2}}{\\Gamma(3/2)\\Gamma(2)} = \\frac{x^{3/2}}{\\sqrt{2\\pi}}$, which matches $\\sqrt{\\frac{2}{\\pi x}} \\cdot \\frac{x^2}{2}$.\nA non-example: $J_{1/2}(x) = \\sqrt{\\frac{2}{\\pi x}} \\sin x$ is not a Struve function; it solves the equation with right-hand side $0$.\nThe modified Struve function is $\\mathbf{L}_\\nu(z) = -i e^{-i\\nu\\pi/2}\\, \\mathbf{H}_\\nu(iz)$.\n",
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