348 



REPORTS ON THE STATE OF SCIENCE. 



exists for any real value of x, say x = c, then it exists for any real value 

 of x greater than c. 



Pincherle, 1 on the other hand, has established the more general 

 theorem that if the integral 







converges for x = «, it must converge for every finite value of x whose 

 real part is greater than or equal to a. Pincherle assumes that <p(t) is 

 finite and continuous in the interval o < t < 1, but this restriction has 

 been removed by Landau. 2 



2. Fourier s Theorems. 



The name of Fourier is greatly honoured among mathematicians for 

 his researches in the conduction of heat 3 wherein he was led to 

 investigate the general problem of the expansion of an arbitrary 

 function in a trigonometrical series, or as it is now called, a Fourier's 

 series. This investigation is only of indirect importance in the theory 

 of integral equations, but his discovery of the inversion formulae 



co 



f(s) = [cos st.'/ (t)dt 



o 



CO 



<f>{t) = - cos st.f(s)ds 





 CO 



g(s) = sin st.\(t)dt 





 CO 



x(t) = - sin st.ij(s)ds 



IT I 



m 

 O 



and of the double integral 



CO CO 



4<(x + o) + i(x— o)— JdAJcos \(x— t)\P(t)dt 



(1) 



(2) 



(3) 



must rank as the greatest discovery in the whole history of the subject. 



The investigations connected with these formulas are very numerous, 4 



and various conditions for their validity have been obtained. The first 



sufficient set of conditions appear to have been given by P. du Bois 



1 Annate* de VEcole normals superieure, 1905 (3), t. 22, pp. 9-68. 



2 See Nielsen's Handbuch dor Gamma FunMioneii, p. 325. 



1 Tkcorie analytiqiie de la chaleur, Paris (1822). Fourier's researches were first 

 presented to the Paris Academy in 1807, and afterwards gained the great mathematical 

 prize in 1812. 



♦ A good bibliography is given in Carslaw's Fourier's Series and Integrals which 

 also contains a proof of the formula (3). Another proof is given in Bromwich's 

 Infinite Series, arts. 19, 20, 169. 



