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  "body": "\n## Definition\n\nLet $J_\\nu$ denote the [[Bessel function of the first kind|CZCMRDWDYW]] of order $\\nu$, taken on $\\C \\setminus (-\\infty, 0]$ with the principal branch of $z^\\nu$.\nFor $\\nu \\in \\C \\setminus \\Z$, the *Bessel function of the second kind* of order $\\nu$ is\n$$Y_\\nu(z) = \\frac{J_\\nu(z) \\cos(\\nu \\pi) - J_{-\\nu}(z)}{\\sin(\\nu \\pi)}.$$\nFor an integer $n$, it is the limit\n$$Y_n(z) = \\lim_{\\nu \\to n} Y_\\nu(z).$$\n\n## Remarks\n\nThe function $Y_\\nu$ solves Bessel's equation $z^2 y'' + z y' + (z^2 - \\nu^2) y = 0$, because it is a linear combination of solutions for non-integer $\\nu$ and a locally uniform limit of solutions for integer $\\nu$.\nIts Wronskian with $J_\\nu$ is $W(J_\\nu, Y_\\nu)(z) = \\frac{2}{\\pi z}$, which never vanishes, so $J_\\nu$ and $Y_\\nu$ are independent for every $\\nu$, integers included.\nThis is the point of the definition: for an integer $n$ one has $J_{-n} = (-1)^n J_n$, so $J_n$ and $J_{-n}$ do not form a basis, while $J_n$ and $Y_n$ do.\nFor $\\nu = \\frac{1}{2}$ the formula gives $Y_{1/2}(x) = -J_{-1/2}(x) = -\\sqrt{\\frac{2}{\\pi x}} \\cos x$ for $x \u003e 0$.\nA non-example: the function $J_{-\\nu}$ alone is not $Y_\\nu$; it is a second solution only when $\\nu \\notin \\Z$.\nThe notation $N_\\nu$ is also used for $Y_\\nu$.\n",
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