97c9e4d2-6de6-4ff1-9d47-f1fba38eb44a 31 4 5 decimal

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The normal Besssel function is not suitable for computation of larger x (here up to 14) since the number range of the computer will be easyly exceeded. John Harris has a very good paper (see left corner) where he decribes the computation of the Bessel function)

x P.0 19128 x 2^* / - 367532768 x 4^* / + 24012454194304 x 6^* / - 130439058752147483648 x 8^* / + :

Here I would need some more sums for the polynoms P0 and Q0.

x Q.0 18 x * / - 751024 x 3^* / + 59535262144 x 5^* / - 5797291533554432 x 7^* / + :

x J.0 2 π x * / sqrt x π 4 / - cos x P.0 * x π 4 / - sin x Q.0 * - (* (:

x J.1 1 - (x J.0 (x diff * :

x J.0 x solve 2.402301 5.520075 8.653728 11.791534 14.930918 18.071064 61 mat

x J.0 x J.1 2 1 sys

x J.1 x solve 3.8315 7.0156 10.1735 13.3237 16.4706 19.6159 61 mat

The zeros look very good so far!

This is the procedure where the author describes how to obtain the sums for P0 and Q0. The more terms the better the accuracy. But maybe I don't understand the notation here --> (n,2m) for Pn(x) respectivly (n,2m+1) for Qn(x)

I don't understand the notation here --> (n,2m) for Pn(x) respectivly (n,2m+1) for Qn(x. This is what I have tried for Pn(x)so far, but have an error here. If somebody would like to help here? The same would be needed for Qn(x) then.

m 1 :

n 0 :

1 - (m^4 n 2^* 12^- (4 n 2^* 32^- (* 4 n 2^* 2 m * 1 - (2^- (* (* 1 * 124 m *^* 12 m * (! (* (* / (2 x * (2 m *^/ 9128 x 2^* /

m 2 :

n 0 :

1 - (m^4 n 2^* 12^- (4 n 2^* 32^- (* 4 n 2^* 52^- (* 4 n 2^* 4 m * 1 - (2^- (* (* 1 * 124 m *^* 12 m * (! (* (* / (2 x * (2 m *^/ 367532768 x 4^* /

m 3 :

n 0 :

1 - (m^4 n 2^* 12^- (4 n 2^* 32^- (* 4 n 2^* 52^- (* 4 n 2^* 72^- (* 4 n 2^* 6 m * 1 - (2^- (* (* 1 * 124 m *^* 12 m * (! (* (* (/ (2 x * (2 m *^/ 708054194304 x 6^* /

m 4 :

n 0 :

1 - (m^4 n 2^* 12^- (4 n 2^* 32^- (* 4 n 2^* 52^- (* 4 n 2^* 72^- (* 4 n 2^* 92^- (* 4 n 2^* 2 m * 1 - (2^- (* (* 24 m *^12 m * (! (* (* / (2 x * (2 m *^/ 1389152147483648 x 8^* /

The pupose of computing the Bessel function is to solve a problem of heat transfer (cylinder)

Bi is the Biot number

he coefficients of the trancendent function (zeros) have been calculated and are summarized in the left table. I want to achive at least this accuracy since the table is from the year 1964.

This is an excerpt from a publication by Prof Grigull. The values for δ had been calculated by computers from Zuse and Siemens in the year 1964 by means of the Newton method. Dr.-Ing. Ulrich Grigull Professor fuer Technische Thermodynamik an der Technischen Hochschule Muenchen Temperaturausgleich in einfachen Koerpern Ebene Platte, Zylinder, Kugel halbunendlicher Koerper Springer Verlag Berlin / Goettingen / Heidelberg 1964 Library of Congress Catalog Card Number 64-15144

This is the trancendent equation which needs to be solved

Biot 110 / :

x Bi x x J.1 * x J.0 / Biot - :

Bi x Bi x solve :

erg x Bi 0.2 105 -^1000 NewtonRaphson :

1 Biot / 10

erg 5 round 32.1928

x Bi

x Bi x 114 solve (3.8446 3.8575 7.009 7.0298 9.9037 10.1833 13.3304 13.3312 81 mat