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  "body": "\n## Definition\n\nLet $0 \u003c k \u003c 1$, $k' = \\sqrt{1 - k^2}$, let $K = K(k)$ be the [[complete elliptic integral of the first kind|RM54CPPE12]], and let $q$ be the [[nome|YSAZVCZAZC]] of $k$.\nFor complex $\\zeta$ let\n$$\\vartheta_1(\\zeta) = 2\\sum_{n \\ge 0} (-1)^n q^{(n + 1/2)^2}\\sin((2n + 1)\\zeta), \\qquad \\vartheta_2(\\zeta) = 2\\sum_{n \\ge 0} q^{(n + 1/2)^2}\\cos((2n + 1)\\zeta),$$\n$$\\vartheta_3(\\zeta) = 1 + 2\\sum_{n \\ge 1} q^{n^2}\\cos(2n\\zeta), \\qquad \\vartheta_4(\\zeta) = 1 + 2\\sum_{n \\ge 1} (-1)^n q^{n^2}\\cos(2n\\zeta)$$\nbe the four Jacobi [[theta functions|2F5CYQJVAF]] with nome $q$.\nWith $\\zeta = \\pi u/(2K)$, the *Neville theta functions* of $u \\in \\C$ are\n$$\\theta_s(u) = \\sqrt{\\frac{\\pi}{2kk'K}}\\,\\vartheta_1(\\zeta), \\quad \\theta_c(u) = \\sqrt{\\frac{\\pi}{2kK}}\\,\\vartheta_2(\\zeta), \\quad \\theta_d(u) = \\sqrt{\\frac{\\pi}{2K}}\\,\\vartheta_3(\\zeta), \\quad \\theta_n(u) = \\sqrt{\\frac{\\pi}{2k'K}}\\,\\vartheta_4(\\zeta).$$\n\n## Remarks\n\nThe constants are chosen so that $\\theta_c(0) = \\theta_d(0) = \\theta_n(0) = 1$ and $\\theta_s'(0) = 1$.\nCheck of $\\theta_d(0) = 1$: Jacobi's formula $\\vartheta_3(0)^2 = 2K/\\pi$ gives $\\theta_d(0) = \\sqrt{\\pi/(2K)}\\cdot\\sqrt{2K/\\pi} = 1$.\nSimilarly $\\vartheta_2(0)^2 = 2kK/\\pi$ and $\\vartheta_4(0)^2 = 2k'K/\\pi$ give $\\theta_c(0) = \\theta_n(0) = 1$.\nThe [[Jacobi elliptic functions|P071B692JF]] are\n$$\\operatorname{sn} u = \\frac{\\theta_s(u)}{\\theta_n(u)}, \\qquad \\operatorname{cn} u = \\frac{\\theta_c(u)}{\\theta_n(u)}, \\qquad \\operatorname{dn} u = \\frac{\\theta_d(u)}{\\theta_n(u)}.$$\nNon-example: $\\vartheta_3(\\pi u/(2K))$ itself is not $\\theta_d(u)$, since its value at $u = 0$ is $\\sqrt{2K/\\pi} \u003e 1$; the factor $\\sqrt{\\pi/(2K)}$ is needed.\nThe function $\\theta_d$ is even with period $2K$, and $\\theta_s$ is odd with $\\theta_s(u + 2K) = -\\theta_s(u)$.\n",
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