The famous transformation formula for the theta function gives 



jj e t . Q ll _ij ( 14 ) 



Substituting (12) into (14), (14) into (13), and (13) into (10) gives 



V4TTfef JO Z- Z^ 



(15) 



Each term represents a concentration pole at S;+2n/ or-£+2rc/, rc = ...., —2, 

 —1, 0, 1, 2, ••••, and thus the initial concentration at § leads to an infinite sequence 

 of concentration poles made up of the primary pole and all its images reflected at 

 z = 0, /. The primary heat pole 



i -(z-V 2 



lAixkt 



could have been deduced initially (it is called the principal solution), and the 

 images could then have been introduced to satisfy the boundary conditions. 



Special Case 



Suppose that initially in the tank there is a distribution of concentration 

 f(£) =0, < 2 < h 



m> = c , h<z<i 



Then (15) becomes 



u(z,t)=-^—f l \\ e -<z-^2nl>>/4kt + y e -f«+g«nWV4*t[ ^ 

 \f4xrkt Jh | Zj /Li 



By letting 



(z+£+2nl) 2 



(16) 



(17) 



(18) 



4kt + 



we transform it into 



Ioc ~ z-k+2 nl oc r z+l+2nl 



y 2 / V4tr e -.j d8+ y_i /vifeT e -s + 2 ds l (19) 



J \f4kT 

 The error function is defined 



2 I Zj VtF / z -l+2nl J— i yfr jz+h+2nl 



erf = ~ f X e" s2 ds (20) 



vu JO 



19 



