

SECT. I.] FUNCTIONS EXPRESSED BY INTEGRALS. 339 



other points are nothing, the function F(x) will be given by this 

 condition. It will then be necessary to integrate, with respect to 

 x, from x to x = 1, for the rest of the integral is nothing 

 according to the hypothesis. We shall thus find 



~ 2 sin q , irv , . C^dg , 2 , 



= ---- * and -TT = e~ M I e q cos qx sm a. 



* 1 - JO 1 I 



^ The second member may easily be converted into a convergent 

 series, as will be seen presently ; it represents exactly the state 

 of the solid at a given instant, and if we make in it t = 0, it ex 

 presses the initial state. 



Thus the function I sin q cos qx is equivalent to unity, if \ 



we give to x any value included between 1 and 1 : but this 

 function is nothing if to x any other value be given not included / 

 between 1 and 1. We see by this that discontinuous functions / / 

 also may be expressed by definite integrals. 



349. In order to give a second application of the preceding 

 formula, let us suppose the bar to have been heated at one of its 

 points by the constant action of the same source of heat, and 

 that it has arrived at its permanent state which is known to be 

 represented by a logarithmic curve. 



It is required to ascertain according to what law the diffusion 

 of heat is effected after the source of heat is withdrawn. Denoting 

 by F (x) the initial value of the temperature, we shall have 



/HL 

 F(x) = Ae A ^; A is the initial temperature of the point 



most heated. To simplify the investigation let us make A = l, 



TTT 



and -^7=1. We have then F(x\e~ x , whence we deduce 

 Ao 



Q = I dx e~ x cos qx, and taking the integral from x nothing to x 



innnite;;&amp;lt; =^j - 3 . T 

 the following equation : 



innnite;;&amp;lt; =^j - 3 . Thus the value of v in x and t is given by 



222 



