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Abstract. In the present paper an attempt is made to improve the series expansion method for computing the incomplete integrals F(0, k) and 77(0, k). Therefore the following three pairs of series covering the region -l5Sfc^l,O^0< ir/2 are used: series obtained by a straightforward binomial expansion of the integrands, series valid for k'2 tan2 0 < 1, and new series which converge for 0 > tt/4 and for all values of k. Terms of the last two pairs of series can be generated by means of the same recurrence relations, so that the coding of the whole is not longer than that for similar methods using only two pairs of series. Any degree of accuracy can be obtained. In general the method is a little bit slower than Bulirsch' calculation procedures which are based on the Landen transformation, but it works more quickly in case of large values of k2 and/or

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Summary. In this paper the problem is investigated of how to take the (possibly noninteger) multiplicity of zeros into account in the Haar condition for a linear function space on a given interval. Therefore, a distinction is made between regular and singular points of the interval, and a notion of geometric multiplicity, which always is a positive integer, is introduced. It is pointed out that, for regular zeros (i.e., zeros situated at regular points), a q-fold zero (in the sense that its geometric multiplicity equals q), counts for q distinct zeros in the Haar condition. For singular zeros (i.e., zeros situated at singular points), this geometric multiplicity has to be diminished by some well-determinable integer.