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BesselYZero[n,k]

represents the k^(th) zero of the Bessel function of the second kind .

BesselYZero[n,k,x0]

represents the k^(th) zero greater than x0.

Details
Details and Options Details and Options
Examples  
Basic Examples  
Scope  
Numerical Evaluation  
Specific Values  
Visualization  
Differentiation and Series Expansions  
Properties & Relations  
See Also
Tech Notes
Related Guides
History
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BesselYZero

BesselYZero[n,k]

represents the k^(th) zero of the Bessel function of the second kind .

BesselYZero[n,k,x0]

represents the k^(th) zero greater than x0.

Details

  • Mathematical function, suitable for both symbolic and numerical manipulation.
  • N[BesselYZero[n,k]] gives a numerical approximation so long as the specified zero exists.
  • BesselYZero[n,k] represents the k^(th) zero greater than 0.
  • BesselYZero can be evaluated to arbitrary numerical precision.
  • BesselYZero automatically threads over lists. »

Examples

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Basic Examples  (5)

Evaluate numerically:

Wolfram Language code: N[BesselYZero[0, 1]]

Evaluate symbolically:

Wolfram Language code: BesselY[0, BesselYZero[0, 1]]

Display zeros of the BesselY function over a subset of the reals:

Wolfram Language code: Plot[BesselY[1, z], {z, 0, 15}, Epilog -> {PointSize[0.03], Red, Point[Table[{BesselYZero[1, k], 0}, {k, 4}]]}]

Series expansion at the origin:

Wolfram Language code: Series[BesselYZero[n, x], {x, 0, 1}]

TraditionalForm formatting:

Wolfram Language code: BesselYZero[ν, k]//TraditionalForm

Scope  (18)

Numerical Evaluation  (7)

Evaluate numerically:

Wolfram Language code: N[BesselYZero[0, 2]]

Find the first zero of greater than 50:

Wolfram Language code: N[BesselJZero[0, 1, 50]]

Evaluate to high precision:

Wolfram Language code: N[BesselYZero[1, 10, 50], 50]

Evaluate efficiently at high precision:

Wolfram Language code: N[BesselYZero[0, 2, 20`100]]//Timing
Wolfram Language code: N[BesselYZero[0, 3, 5`1000]];//Timing

Evaluate at a noninteger second argument:

Wolfram Language code: N[BesselYZero[0, 2 - 2 / 3], 20]

For BesselYZero[ν,k-α/π], the result is a zero of :

Wolfram Language code: BesselJ[0, %]Sin[2Pi / 3] + BesselY[0, %]Cos[2Pi / 3]

Compute the elementwise values of an array using automatic threading:

Wolfram Language code: BesselYZero[1 / 2, {{2, 1}, {1, 3}}]//N

Or compute the matrix BesselYZero function using MatrixFunction:

Wolfram Language code: MatrixFunction[BesselYZero[1 / 2, #]&, {{2, 1}, {1, 3}}]//N

Compute average-case statistical intervals using Around:

Wolfram Language code: BesselYZero[ 1, Around[2, 0.01]]

Specific Values  (3)

Limiting value at infinity:

Wolfram Language code: Limit[BesselYZero[2, x], x -> Infinity]

The first three zeros:

Wolfram Language code: {BesselYZero[0, 1], BesselYZero[0, 2], BesselYZero[0, 3]}//N

Find the first zero of BesselY[1,x] using Solve:

Wolfram Language code: xzero = x /. Solve[BesselY[1, x] == 0 && 2 < x < 6, x][[1]]
Wolfram Language code: Plot[BesselY[1, x], {x, 0, 10}, Epilog -> Style[Point[{xzero, BesselY[1, xzero]}], PointSize[Large], Red]]

Visualization  (3)

Visualize the zeroes of BesselY as a step function:

Wolfram Language code: Plot[BesselYZero[0, 1, x], {x, 1, 10}]

Display zeros of the BesselY function:

Wolfram Language code: Plot[BesselY[1, z], {z, 0, 20}, Epilog -> {PointSize[0.03], Point[Table[{BesselYZero[1, k], 0}, {k, 6}]]}]

Show the first zero greater than 4:

Wolfram Language code: Plot[BesselY[1, z], {z, 0, 10}, Epilog -> {PointSize[0.03], Red, Point[{BesselYZero[1, 1, 4], 0}]}]

Differentiation and Series Expansions  (5)

Derivative of Bessel zero with respect to k:

Wolfram Language code: D[BesselYZero[ν, k], k]

Second derivative:

Wolfram Language code: D[BesselYZero[2, x], {x, 2}]// FullSimplify

Find the Taylor expansion using Series:

Wolfram Language code: Series[BesselYZero[n, x], {x, 0, 2}]//Normal// Simplify

Find the series expansion at Infinity:

Wolfram Language code: Series[BesselYZero[n, x], {x, Infinity, 1}]

Taylor expansion at a generic point:

Wolfram Language code: Series[BesselYZero[n, x], {x, x0, 1}]

Properties & Relations  (1)

Asymptotic behavior of BesselYZero[ν,k] for large k:

Wolfram Language code: Series[BesselYZero[ν, k], {k, Infinity, 3}]
Wolfram Research (2007), BesselYZero, Wolfram Language function, https://reference.wolfram.com/language/ref/BesselYZero.html.

Text

Wolfram Research (2007), BesselYZero, Wolfram Language function, https://reference.wolfram.com/language/ref/BesselYZero.html.

CMS

Wolfram Language. 2007. "BesselYZero." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/BesselYZero.html.

APA

Wolfram Language. (2007). BesselYZero. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BesselYZero.html

BibTeX

@misc{reference.wolfram_2026_besselyzero, author="Wolfram Research", title="{BesselYZero}", year="2007", howpublished="\url{https://reference.wolfram.com/language/ref/BesselYZero.html}", note=[Accessed: 13-June-2026]}

BibLaTeX

@online{reference.wolfram_2026_besselyzero, organization={Wolfram Research}, title={BesselYZero}, year={2007}, url={https://reference.wolfram.com/language/ref/BesselYZero.html}, note=[Accessed: 13-June-2026]}