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Bessel Function Zeros


BesselFunctionZeros

When the index nu is real, the functions J_nu(z), J_nu^'(z), Y_nu(z), and Y_nu^'(z) each have an infinite number of real zeros, all of which are simple with the possible exception of z=0. For nonnegative nu, the kth positive zeros of these functions are denoted j_(nu,k), j_(nu,k)^', y_(nu,k), and y_(nu,k)^', respectively, except that z=0 is typically counted as the first zero of J_0^'(z) (Abramowitz and Stegun 1972, p. 370).

The first few roots j_(n,k) of the Bessel function J_n(x) are given in the following table for small nonnegative integer values of n and k. They can be found in the Wolfram Language using the command BesselJZero[n, k].

kJ_0(x)J_1(x)J_2(x)J_3(x)J_4(x)J_5(x)
12.40483.83175.13566.38027.58838.7715
25.52017.01568.41729.761011.064712.3386
38.653710.173511.619813.015214.372515.7002
411.791513.323714.796016.223517.616018.9801
514.930916.470617.959819.409420.826922.2178

The first few roots j_(n,k)^' of the derivative of the Bessel function J_n^'(x) are given in the following table for small nonnegative integer values of n and k. Versions of the Wolfram Language prior to 6 implemented these zeros as BesselJPrimeZeros[n, k] in the BesselZeros package which is now available for separate download (Wolfram Research). Note that contrary to Abramowitz and Stegun (1972, p. 370), the Wolfram Language defines the first zero of J_0^'(z) to be approximately 3.8317 rather than zero.

kJ_0^'(x)J_1^'(x)J_2^'(x)J_3^'(x)J_4^'(x)J_5^'(x)
13.83171.84123.05424.20125.31756.4156
27.01565.33146.70618.01529.282410.5199
310.17358.53639.969511.345912.681913.9872
413.323711.706013.170414.585815.964117.3128
516.470614.863616.347517.788719.196020.5755

See also

Bessel Function, Bessel Function of the First Kind

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References

Abramowitz, M. and Stegun, I. A. (Eds.). "Zeros." §9.5 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 370-374, 1972.Goodwin, E. T. and Staton, J. "Table of J_0(j_(0,n)r)." Quart. J. Mech. Appl. Math. 1, 220-224, 1948.Olver, F. W. J. (Ed.). "Zeros and Associated Values." Royal Society Mathematical Tables, Vol. 7: Bessel Functions. Cambridge, England: Cambridge University Press, 1960.Wolfram Research. "Wolfram Language & System Documentation Center: Upgrading from NumericalMath BesselZeros." http://reference.wolfram.com/language/Compatibility/tutorial/NumericalMath/BesselZeros.html.Wolfram Research. "Wolfram Library Archive: NumericalMath BesselZeros Legacy Standard Add-On Package." library.wolfram.com/infocenter/MathSource/6777.

Referenced on Wolfram|Alpha

Bessel Function Zeros

Cite this as:

Weisstein, Eric W. "Bessel Function Zeros." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BesselFunctionZeros.html

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