Logarithmic, Circular, Exponential Functions 



4030 



Potential expression ((!))*. Kinkelin, H. 



Grunert Arch. 26 (1856) 304- . 

 Eeciprocal series of sines and cosines of arcs in 



arithmetical progression. Lorgna, A. M. 



Turin Mm. Ac. 3 (1786-87) 215-. 

 Series formed from series for log(l + :r) etc. 



Hellwig, C. Grunert Arch. 21 (1853) 43-. 

 - for roots and logarithms. Bouvier, L. C. 



Gergonne A. Mth. 16 (1825-26) 294-. 

 l_^-)-.i J..., note. Lionnet, E. N. A. 



Mth. 18 (1879) 509-. 

 Stirling's approximation to n ! Allardice, E. E. 



Edinb. Mth. S. P. 6 (1888) 22-. 



formula, proof by logarithmic inequalities. 

 Mansion, P. Mathesis 20 (1900) 265-. 



theorem, elementary proof. Franklin, F. 

 J. H. Un. Cir. [3] (1884) 57. 



Transcendental functions arising from repeated 

 integration of rational functions. Jonquiere, 

 A. Stockh. Ak. Hndl. Bh. 15 (Afd. 1) (1890) 

 No. 1, 50pp. 



, class, integration. Winckler, A. [1874] 



Wien Ak. Sb. 70 (1875) (Ab. 2) 17-. 



integrals of form / e~ x ^fxdx. Thomson, 



(Sir) W. QJ. Mth. 1 (1857) 316-. 

 Wronski, formula, 



4030 



I" I 



+ constant, 



\F(x)dx = l{W 



where 



1 dFx 1 d?Fx ~\ 

 ^ x=Fx+ _.__ + __. ^- + ...J. 



Rodrigues, J. M. G. Teix. J. Sc. 3 (1881) 

 55-. 



\e nx x m X 1 X^X^dx, where X^^ are poly- 

 nomials in x. Melanderhjelm, D. [1798] 

 St Pet. Ac. Sc. N. Acta 12 (1801) 114-. 



to* ' Bidme ' G ' Turin Mm ' Ac ' ( 1805 - 8 ) 

 (pte. 2) 19-. 



-. Caluso, (Vabbe) T. V. de. Mod. Mm. S. 

 It. 12 (1805) 268-. 



1^. Bessel,F. W. Zach M. Cor. 22 (1810) 



395-; Konigsb. Arch. 1 (1812) 1-, 239-. 

 . Buzengeiger, K. H. J. Zach M. Cor. 28 



(1813) 373-. 

 . Casinelli, A. (s. L.). Bologna N. Cm. 3 



(1839) 97-. 

 x x =A, solution. Leyonmark, G. A. Stockh. 



Ak. Hndl. 31 (1810) 161-. 

 x xn . Carlini, F. Mil. Mm. I. Lomb. Yen. 



1 (1812-13) 167-. 

 e 2 **, trigonometric expression for. Busse, F. G. 



von. Oken Isis (1825) 394-, 528-. 

 log ( - x) and Bouvier. Busse, F. G. von. 



Oken Isis (1825) 1048-. 

 0*. Libri, G. Crelle J. 6 (1830) 67-. 

 ir=?logi. Schellbach, C. H. Crelle J. 9 



(1832) 404-. 



le x ydx, integration where y is algebraic. 



Liouville, J. Crelle J. 13 (1835) 93-. 

 log (1 + x) m . Casinelli, A. (s. L.). Bologna N. 

 Cm. 4 (1840) 147-. 



^ , limiting value for x=<x>. Schulten, 

 N. G. af (fil.) Helsingf. Acta 1 (1840) 



logo; and a x , differentials. Grunert, J. A. 



Grunert Arch. 1 (1841) 204-. 

 log a; dx, higher integrals. Arndt, F. Grunert 



Arch. 4 (1844) 436-. 



e v ~ 19 , use in transformations. Goodwin, H. 

 Camb. Mth. J. 4 (1845) 177-. 



expansion. Meyer, U. H. 



Grunert Arch. 9 (1847) 101-. 

 e 31 , expansion in series. Brassinne, E. N. A. 



Mth. 7 (1848) 30-. 

 a x >x, condition. Hessel, J. F. C. Grunert 



Arch. 14 (1850) 93-. 

 xv and log (x). Bjorling, E. G. Stockh. Ak. 



Hndl. (1852) 121-. 



Pr^^' 



f^ dx 2 , /C=, / f****- 



te 



J r, , 



Roberts, W. Liouv. J. Mth. 17 (1852) 



117-. 

 x a and a x , differentials, calculation. Schlomilch, 



O. N. A. Mth. 12 (1853) 31-. 

 log x and tan ~ l x, differentials, and expansions 



in series. SouMet, . N. A. Mth. 12 (1853) 



438-. 

 Iog e 2, Iog e 3, Iog e 5, log, 10 to 205 places 



(Napierian). Shanks, W. [1854-71] E. S. 



P. 6 (1854) 397-; 20 (1872) 27-. 



" ' = JL [go H . e -a _ e -an . e a^ 



Zehfuss, G. Grunert Arch. 30 (1858) 465-. 

 e+w. Dienger, J. Grunert Arch. 33 (1859) 



481. 

 {(*)}, log {(*)}, theory. Almqvist, P. W. 



Stockh. Ofv. 20 (1863) 203-. 



y = e* x , representation in form y = S[A m e mx ]. 



Spitzer, S. Arch. Mth. Ps. 42 (1864) 



104-. 

 e x , limits. Steen, A. (xn) Ts. Mth. 3 (1867) 



29-. 



f f (x\ 



I e* J ~Jr , integration. Tissot, A . Par. EC. Pol. 

 J V x 



J. 43 e cah. (1870) 201-. 

 y = x n e* x *, representation in form y = S [A m e*]. 



Spitzer, S. Arch. Mth. Ps. 52 (1871) 364-. 

 y = x n e alfi , representation in form y = S [A m e" 1 *]. 



Spitzer, S. Arch. Mth. Ps. 52 (1871) 368-. 

 log M = log . log e, value to 24 places, also 



log Mir. Gray, P. Mess. Mth. 5 (1876) 



172-. 

 x n e* x *, transformation. Spitzer, S. Arch. 



Mth. Ps. 58 (1876) 431-. 

 e', period. Yvon-Villarceau, A. J. F. C. E. 



83 (1876) 594-. 



310 



