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  "body": "\n## Definition\n\nLet $\\nu \\in \\C$, let $\\Gamma$ be the [[gamma function|0H5SYBSED9]], and let $z \\in \\C \\setminus (-\\infty, 0]$ with the principal branch of $z^\\nu$.\n\n1. The *modified Bessel function of the first kind* of order $\\nu$ is\n$$I_\\nu(z) = \\sum_{m=0}^{\\infty} \\frac{1}{m!\\, \\Gamma(m + \\nu + 1)} \\left( \\frac{z}{2} \\right)^{2m + \\nu},$$\nwhere $\\frac{1}{\\Gamma(m + \\nu + 1)}$ is read as $0$ when $m + \\nu + 1$ is a non-positive integer.\n2. For $\\nu \\notin \\Z$, the *modified Bessel function of the second kind* of order $\\nu$ is\n$$K_\\nu(z) = \\frac{\\pi}{2}\\, \\frac{I_{-\\nu}(z) - I_\\nu(z)}{\\sin(\\nu \\pi)},$$\nand for an integer $n$ it is $K_n(z) = \\lim_{\\nu \\to n} K_\\nu(z)$.\n\n## Remarks\n\nBoth functions solve the modified Bessel equation $z^2 y'' + z y' - (z^2 + \\nu^2) y = 0$.\nIn terms of the [[Bessel function of the first kind|CZCMRDWDYW]], $I_\\nu(z) = i^{-\\nu} J_\\nu(iz)$ for $-\\pi \u003c \\arg z \\le \\frac{\\pi}{2}$.\nExample: for $x \u003e 0$, $I_{1/2}(x) = \\sqrt{\\frac{2}{\\pi x}} \\sinh x$ and $K_{1/2}(x) = \\sqrt{\\frac{\\pi}{2x}}\\, e^{-x}$.\nFor $x \u003e 0$ there is the integral form $K_\\nu(x) = \\int_0^\\infty e^{-x \\cosh t} \\cosh(\\nu t)\\, dt$.\nA non-example: $I_{-n}$ for an integer $n$ is not a second solution, since $I_{-n} = I_n$; this is why $K_n$ is defined as a limit.\n$K_\\nu$ is also called the Macdonald function.\n",
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