Bessel functions of the first kind

${J}_{n}\left(z\right)$ and the second kind

${Y}_{n}\left(z\right)$ of integer order play an important role in mathematical physics and engineering sciences. Numerical methods for efficiently computing these functions are therefore of interest to computational physicists and engineers. The authors employ the truncated Chebychev series to approximate the Bessel function of the second kind

${Y}_{n}\left(z\right)$ for

$\left|z\right|\le 8$. Detailed manipulations and discussions for

${Y}_{0}\left(z\right)$ and

${Y}_{1}\left(z\right)$ are given. Results of numerical experiments are presented to demonstrate the computed accuracy by using the Chevychev series approximation. The computed accuracy is comparable with that computed by the tau-method approximations, especially when

$argz$ is small. Advantages and disadvantages of the Chebychev series approximation compared with the tau-method approximation are discussed.