Geodesic
Approaches to the summability of divergent multidimensional integrals
Sbornik. Mathematics, Tome 194 (2003) no. 8, pp. 1137-1166 
G. M. Vainikko; I. K. Lifanov. Approaches to the summability of divergent multidimensional integrals. Sbornik. Mathematics, Tome 194 (2003) no. 8, pp. 1137-1166. http://geodesic.mathdoc.fr/item/SM_2003_194_8_a2/
@article{SM_2003_194_8_a2,
     author = {G. M. Vainikko and I. K. Lifanov},
     title = {Approaches to the summability of divergent multidimensional integrals},
     journal = {Sbornik. Mathematics},
     pages = {1137--1166},
     year = {2003},
     volume = {194},
     number = {8},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2003_194_8_a2/}
}
TY  - JOUR
AU  - G. M. Vainikko
AU  - I. K. Lifanov
TI  - Approaches to the summability of divergent multidimensional integrals
JO  - Sbornik. Mathematics
PY  - 2003
SP  - 1137
EP  - 1166
VL  - 194
IS  - 8
UR  - http://geodesic.mathdoc.fr/item/SM_2003_194_8_a2/
LA  - en
ID  - SM_2003_194_8_a2
ER  - 
%0 Journal Article
%A G. M. Vainikko
%A I. K. Lifanov
%T Approaches to the summability of divergent multidimensional integrals
%J Sbornik. Mathematics
%D 2003
%P 1137-1166
%V 194
%N 8
%U http://geodesic.mathdoc.fr/item/SM_2003_194_8_a2/
%G en
%F SM_2003_194_8_a2

Voir la notice de l'article provenant de la source Math-Net.Ru

Under discussion are various approaches to the concept of summability (finding the finite part – f.p. ) of divergent integrals with integrand represented as a product of two functions, one with a parameter-dependent non-integrable singularity at one point of the integration and the other absolutely integrable. A study is made of summability methods which are based on the expansion of the absolutely integrable function in a Taylor series with centre at the singular point (f.p.), on the analytic continuation with respect to the parameter of the singularity (a.f.p.), and on integration by parts (f.p.p.). Formulae of changes of variables in such integrals are presented.

[1] Muskelishvili N. I., Singulyarnye integralnye uravneniya, Nauka, M., 1968 | MR

[2] Gakhov F. D., Kraevye zadachi, Nauka, M., 1977 | MR

[3] Belotserkovsky S. M., Lifanov I. K., Method of discrete vortices, CRC Press, 1993 | MR

[4] Lifanov I. K., Metod singulyarnykh integralnykh uravnenii i chislennyi eksperiment, TOO “Yanus”, M., 1995 | MR | Zbl

[5] Lifanov I. K., Singular integral eguations and discrete vortices, VSP, Utrecht, 1996 | MR | Zbl

[6] Zakharov E. V., Pimenov Yu. V., Chislennyi analiz difraktsii radiovoln, Radio i svyaz, M., 1982 | MR | Zbl

[7] Parton V. Z., Perlin P. I., Metody matematicheskoi teorii uprugosti, Nauka, M., 1981 | MR

[8] Saranen J., Vainikko G., Periodic integral and pseudodifferential equations with numerical approximation, Springer-Verlag, Berlin, 2002 | MR | Zbl

[9] Vainikko G. M., Lifanov I. K., “O ponyatii konechnoi chasti raskhodyaschikhsya integralov”, Differents. uravneniya, 38:9 (2002), 1233–1246 | MR | Zbl

[10] Vainikko G. M., Lifanov I. K., Poltavskii L. N., Chislennye metody v gipersingulyarnykh integralnykh uravneniyakh, TOO “Yanus-K”, M., 2001

[11] Mikhlin S. G., Mnogomernye singulyarnye integraly i integralnye uravneniya, Fizmatgiz, M., 1962 | MR | Zbl

[12] Anfinogenov A. Yu., Lifanov I. K., Lifanov P. I., “O nekotorykh odno- i dvumernykh gipersingulyarnykh integralnykh uravneniyakh”, Matem. sb., 192:8 (2001), 3–46 | MR | Zbl