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{
"Title": "Values of the Kelvin function $\\operatorname{bei}_\\nu(x)$",
"Definition": "The Kelvin function $\\operatorname{bei}_\\nu(x)$ is defined by $\\operatorname{bei}_\\nu(x)=\\Im J_\\nu(xe^{3\\pi i/4})$ CITE{DLMF}.",
"Keywords": [
"Kelvin bei",
"Kelvin function",
"bei0",
"bei1",
"bei2",
"cylinder functions"
],
"Parameters": {
"nu": {
"type": "Q",
"title": "order",
"display": "$\\nu$",
"constraints": "$\\nu\\geq0$"
},
"x": {
"type": "Q",
"title": "argument",
"display": "$x$",
"constraints": "$x>0$"
}
},
"Comments": {
"comment-branch": "$J_\\nu$ denotes the principal branch of HREF{Values_of_Bessel_functions_of_the_first_kind}[the Bessel function of the first kind]. The rotated argument has modulus $x$ and angle $3\\pi/4$ CITE{DLMF}.",
"comment-companion": "HREF{T490}[The Kelvin function $\\operatorname{ber}_\\nu(x)$] is the real part of the same rotated Bessel value. The notation $\\operatorname{bei}(x)$ without an order means $\\operatorname{bei}_0(x)$ CITE{DLMF}.",
"comment-origin": "The continuous value at the origin is $\\operatorname{bei}_\\nu(0)=0$ for $\\nu\\geq0$, as follows from the Bessel power series CITE{DLMFSeries}.",
"comment-integer-orders": "At integer order the defining Bessel function is entire, so no branch choice remains. CITE{formula-reflections} extends the listed integer-order values to negative orders and negative real arguments."
},
"Formulas": {
"formula-rotation": "$\\operatorname{bei}_\\nu(x)=\\Im\\!\\left(e^{\\nu\\pi i/2}I_\\nu(xe^{\\pi i/4})\\right)$, where $I_\\nu$ is HREF{Values_of_modified_Bessel_functions_of_the_first_kind}[the modified Bessel function of the first kind] CITE{DLMF}.",
"formula-series": "For an integer $n\\geq0$, $\\operatorname{bei}_n(x)=\\left(\\dfrac{x}{2}\\right)^n\\sum_{k=0}^{\\infty}\\dfrac{\\sin(3n\\pi/4+k\\pi/2)}{k!(n+k)!}\\left(\\dfrac{x^2}{4}\\right)^k$ CITE{DLMFSeries}.",
"formula-recurrence": "$\\operatorname{bei}_{\\nu-1}(x)+\\operatorname{bei}_{\\nu+1}(x)=-\\dfrac{\\nu\\sqrt{2}}{x}\\bigl(\\operatorname{bei}_\\nu(x)+\\operatorname{ber}_\\nu(x)\\bigr)$ for $x>0$ and $\\nu\\geq1$, where HREF{T490}[$\\operatorname{ber}_\\nu(x)$] is the companion Kelvin function CITE{DLMFRecurrences}.",
"formula-reflections": "For an integer $n\\geq0$, $\\operatorname{bei}_{-n}(x)=(-1)^n\\operatorname{bei}_n(x)$ and $\\operatorname{bei}_n(-x)=(-1)^n\\operatorname{bei}_n(x)$ CITE{DLMF}."
},
"Programs": {
"program-sage": {
"language": "Sage",
"code": "import numberdb.sage as numberdb\nfrom sage.rings.complex_arb import ComplexBallField\nfrom sage.rings.rational_field import QQ\n\nC = ComplexBallField(numberdb.bits(100, losing=64))\nnu, x = QQ(0), QQ(1)\nz = C(x) * C(-1, 1) / C(2).sqrt()\nvalue = z.bessel_J(C(nu)).imag()\nprint(value)"
},
"program-python": {
"language": "Python",
"code": "import mpmath as mp\nmp.mp.dps = 100\nvalue = mp.bei(0, 1)\nprint(value)"
}
},
"Similar tables": [
{
"table": "HREF{T490}",
"relation": "$\\operatorname{ber}_\\nu$ and $\\operatorname{bei}_\\nu$ are the real and imaginary parts of the same rotated Bessel value"
},
{
"table": "HREF{Values_of_Bessel_functions_of_the_first_kind}",
"relation": "$\\operatorname{bei}_\\nu$ is the imaginary part of $J_\\nu$ after rotation of its argument by $3\\pi/4$"
},
{
"table": "HREF{Values_of_modified_Bessel_functions_of_the_first_kind}",
"relation": "the rotated $I_\\nu$ representation in CITE{formula-rotation} gives the same Kelvin function"
},
{
"table": "HREF{Values_of_Bessel_functions_of_the_second_kind}",
"relation": "$Y_\\nu$ is the second standard solution of the Bessel equation whose rotated solutions give the Kelvin functions"
},
{
"table": "HREF{Values_of_modified_Bessel_functions_of_the_second_kind}",
"relation": "rotating $K_\\nu$ gives the Kelvin pair $\\operatorname{ker}_\\nu,\\operatorname{kei}_\\nu$, while rotating $I_\\nu$ gives $\\operatorname{ber}_\\nu,\\operatorname{bei}_\\nu$"
}
],
"Links": {
"DLMF": {
"title": "DLMF 10.61: Definitions and Basic Properties",
"url": "https://dlmf.nist.gov/10.61"
},
"DLMFRecurrences": {
"title": "DLMF 10.63: Recurrence Relations and Derivatives",
"url": "https://dlmf.nist.gov/10.63"
},
"DLMFSeries": {
"title": "DLMF 10.65: Power Series",
"url": "https://dlmf.nist.gov/10.65"
},
"DLMFIntegral": {
"title": "DLMF 10.9: Integral Representations",
"url": "https://dlmf.nist.gov/10.9#E2"
}
},
"Tags": [
"special functions",
"special values"
],
"Data properties": {
"type": "R",
"digits": 100,
"rigour": "proven",
"complete": "no",
"complete-note": "it holds orders $\\nu=0,1,2$ at every positive rational $x=a/b$ in lowest terms with $b\\leq4$ and $x\\leq20$; these small denominators select simple fractional arguments, and the range includes the small-argument regime and several sign changes of the low-order functions",
"rigour details": "Each value is the imaginary part of a Sage complex-ball evaluation of the defining Bessel function. The rotation is computed as $(-1+i)/\\sqrt{2}$ in ball arithmetic, which also encloses rounding of the rational argument. The working precision is `numberdb.bits(100, losing=64)`, or 397 bits. Across the 360 values, every returned ball is finite, the largest radius is less than $2.66\\cdot10^{-113}$, and every relative radius is less than $5.62\\cdot10^{-115}$. Each ball lies inside the interval denoted by its stored 100-significant-digit decimal.\n\nIndependent 600-bit enclosures from CITE{formula-series} use exact rational coefficients and the exact sine cycle, with a ball enclosure of $1/\\sqrt{2}$ for odd order. The sum uses 200 terms. With $a_k=(x/2)^n(x^2/4)^k/(k!(n+k)!)$ and $r=(x^2/4)/(201(n+201))$, the absolute omitted tail is bounded by $a_{200}/(1-r)$. This is less than $1.62\\cdot10^{-350}$ throughout the table. Every written interval contains this independent enclosure.\n\nThe attached checks also compare every value with mpmath at 160 decimal digits and with CITE{formula-rotation} at 600 bits. Numerical quadrature of Bessel's integral CITE{DLMFIntegral}, using elementary functions at 125 decimal digits, agrees to at least 110 relative decimal places on every row. Before these measurements, the integral is tested at $J_0(0)=1$ and $J_1(0)=J_2(0)=0$; the independent series is tested at $\\operatorname{ber}_0(0)=1$ and $\\operatorname{bei}_0(0)=0$. These numerical comparisons supplement the certified enclosures.\n\nCITE{formula-reflections} is tested on every row. CITE{formula-recurrence} is tested at all 120 arguments with $\\nu=1$, where all three orders are present. The checks can repeat all comparisons on the written intervals read from the database and use the stored companion values from HREF{T490}[the table of $\\operatorname{ber}_\\nu(x)$] in the recurrence."
},
"Display properties": {
"number-header": "$\\operatorname{bei}_\\nu(x)$"
}
}