4040 



Elliptic and Single Theta Functions 



4040 



log(l + x), expansion. Fuhrmann, W. Arch. 



Mth. Ps. 62 (1878) 220. 

 r* _z* 

 I z n e 2 zx dz. Laguerre,E. [1878] Par. S. 



Mth. Bll. 7 (1879) 12-. 



log 2, &c., calculation (Gauss). Cayley, A. 

 Mess. Mth. 8 (1879) 125-. 



I fL^: 

 Z 



. Laguerre, E. Par. S. Mth. Bll. 7 

 (1879) 72-. 



I \ogsinxdx, determination. Ligowski, W. 



[1879] Arch. Mth. Ps. 65 (1880) 110-. 

 F and (-!)*. Laisant, C. A. Par. S. Mth. 



Bll. 8 (1880) 109-. 

 e*, log e (lx), (l + x) m , series for. HOI, 



M. J. M. Camb. Ph. S. P. 5 (1886) 415-. 

 e az :(e z -\), resolution into partial fractions. 



Stolz, O. Innsb. Nt. Md. B. 15 (1886) 



84-. 

 COSP x sin" x dx, integration. Laisant, C. A. 



As. Fr. C. E. (1889) (Pt. 2) 225-. 

 e x , regular combined fractions for. Fade, H. 



C. E. 112 (1891) 712-. 

 log x and e*, independent definitions. McClin- 



tock, E. Am. J. Mth. 14 (1892) 72-. 



I cos yx dxl(a 2 + b 2 x y ). Oltramare, G. As. 

 Fr. C. E. (1895) (Pt. 2) 167-. 



, expression by logarithms. 



Dolbnia, I. P. Bll. Sc. Mth. 19 (1895) 76- ; 

 Eec. Mth. (Moscou) 18 (1896) 108-. 

 e^), series for. Petrovitch, M. Palermo 

 Cir. Mt. Ed. 14 (1900) 22-. 



4040 General properties of elliptic 

 functions and single theta func- 

 tions ; addition-theorem. (See 

 also 8050, 8060.) 



Addition. 



of arguments in elliptic functions. Hermite, C. 



G. Teix. J. Sc. 11 (1892) 65-. 

 periodic functions, 2nd order. Fontene, 



G. C. E. 122 (1896) 172-. 



elliptic functions. Cayley, A. Crelle J. 

 41 (1851) 57-. 



. Darboux, G. Par. EC. Norm. A. 4 



(1867) 81-. 

 . Catalan, E. C. C. E. 78 (1874) 



1479-. 



. Frobenius, G., (& Stickelberger, L. 

 [1879] Crelle J. Mth. 88 (1880) 146-. 



. Cayley, A. Mess. Mth. 14 (1885) 



56-. 

 , formula. Gilbert, L. P. (xii) Brux. 



S. Sc. A. 4 (1880) (Pt. 2) 87-. 

 , formulae. Bjorling, E. G. [1866] 



Arch. Mth. Ps. 47 (1867) 399-. 



of elliptic functions, formulae of Hermite. 



Novarese, E. [1882] Tor. Ac. Sc. At. 17 



(1881) 607-. 

 , Jacobi. Broch, 0. J. C. E. 59 



(1864) 999-. 

 , 1st kind. Catalan, E. C. Brux. Ac. 



Bll. 27 (1869) 145-. 

 , . Catalan, E. Brux. Ac. Mm. 



45 (1884) (No. 3) 20 pp. 

 , , and Euler's integral. laggi, 



E. N. A. Mth. 19 (1900) 443-. 

 integrals, geometrical application. Hoppe, 



E. Arch. Mth. Ps. 64 (1879) 274-. 

 equation. Thomson, J. J. [1879] Mess. Mth. 



9 (1880) 52-. 



for elliptic integrals, 1st kind, geometric 

 construction. Hermite, C. Bll. Sc. Mth. 

 As. 2 (1871) 21-. 



, 2nd kind, proof by ^-series. 

 GlaisJier, J. W. L. [1882] Mess. Mth. 12 

 (1883) 43-. 



, 3rd kind. Glaisher, J. W. L. 



[1880-81] Mess. Mth. 10 (1881) 124-. 

 , spherical triangle proof. Johnson, W. W. 



QJ. Mth. 18 (1882) 365-. 

 equations for elliptic and theta functions of 



sum of n arguments. Wilkinson, M. M. U. 



L. Mth. S. P. 13 (1881-82) 110-. 



ADDITION-THEOBEM. 



Story, W. E. Am. J. Mth. 7 (1885) 364-. 

 Mukhopadhyay, A. QJ. Mth. 21 (1886) 



212-. 

 Giinther, P. Crelle J. Mth. 109 (1892) 



213-. 

 Teixeira, J. P. G. Teix. J. Sc. 11 (1892) 136-; 



Fschr. Mth. (1893-94) 767. 

 for elliptic integrals. Olsson, 0. Ts. Mth. 5 



(1887) 33-; Fschr. Mth. (1887) 455. 

 , construction, geometric. Morera, G. 



[1880] Tor. Ac. Sc. At. 15 (1879) 649-. 



, demonstration. Gwyther, E. F. 

 Manch. Lt. Ph. S. P. 22 (1883) 81-. 



, derivation from equation + -rr = 0. 



Graefe, F. [1880] Crelle J. Mth. 90 (1881) 



83-. 

 , Euler's. Eichelot, F. J. Crelle J. 



44 (1852) 277-. 

 , , Degen's attempt to generalise. 



Bjerknes, C. A. Bb. Mth. (1888) 1-. 



, 1st kind, demonstration, Sturm's. 

 SehriJter, H. Z. Mth. Ps. 17 (1872) 508-. 



, , demonstrations. Bigler, U. 



Arch. Mth. Ps. 7 (1889) 401-. 

 . , , derivation, method of Sturm. 



Much, . Z. Mth. Ps. 26 (1881) 333-. 

 , , 2nd and 3rd kinds. Dolbnia, I. 



Kazan S. Nt. (Ps.-Mth.) P. 7 (1889) 364-; 



N. A. Mth. 8 (1889) 204-. 

 , 2nd kind, demonstrations. Walker, 



J. J. L. Mth. S. P. 13 (1881-82) 172-. 

 , and 3rd kinds. Forsyth, A. B. 



Mess. Mth. 15 (1886) 49-. 

 , theory. Schellbach, C. H. Crelle J. 



54 (1857) 59-. 



311 



