ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

11. Mathematical Physics

1991 MS Classification: 44, 65 Perovich, S. M., University of Montenegro, Montenegro Some Comments Concerning the Application of the Discrete Eigenvalue Theory The analytical solution to the integral of the form

J(c) + ∫_{0}^{1} arctg \{\frac{Cπx}{2(1−cπarctgh(x))}\} xdx

is obtained, by application of the Transcendental Method Theory [1] and Discrete Eigenvalue Theory [2], in the form J(c) = \frac{3−2c}{3−3c} −((disperz(c)))^{2}, where disperz(c) is the novel special function defined as

disperz(c(z)) = lim_{y→∞} \{In [ \frac{\frac{∂}{∂y} (N(z,y−f(z))exp(y−f(z)))}{\frac{∂}{∂y} (N(z,y)exp(y))}]\}

where

N(z, y) = ∑_{n=0}^{[y/f(z)]} ∑_{k=0}^{n}(−1)^{n}e^{−nf(z)} \frac{2^{k}(y−nf(z))^{k}}{(n−k)!k!k!} , f = 2/c.

Or, more explicitly, we have disperz(c) =

\frac{∑_{n=0}^{N−1} ∑_{k=0}^{n}(−1)^{n}e^{−nf} \frac{(2f)^{k}(N−1−n)^{k}n!}{(n−k)!k!k!} + ∑_{n=0}^{N−1} ∑_{k=0}^{n}(−1)^{n}e^{−nf} \frac{(2f)^{k}(N−1−n)^{k}n!}{(n−k)!(k−1)!k!}+}{∑_{n=0}^{N} ∑_{k=0}^{n}(−1)^{n}e^{−nf} \frac{(2f)^{k}(N−n)^{k}n!}{(n−k)!k!k!} + ∑_{n=0}^{N} ∑_{k=0}^{n}(−1)^{n}e^{−nf} \frac{(2f)^{k}(N−n)^{k}n!}{(n−k)!(k−1)!k!}+}

References: [1] Perovich, S. M. Transport Theory and Statistical Physics, 1997, 26, 705. [2] Siewert, C. E., Journal of Mathematical Physics, 1980, 21, 2468.