EXPANSION OF FUNCTIONS IN SERIES. 



For put x + h = z. 



In the first case 



= df(z] dz 

 dx dz ' dx 



M , \ dz 

 =/ (z), since ^- = 1. 



In the second case, 



df(x + h) = df(z) dz_ ' 

 dh dz ' dh 



/., . . . dz 

 =f (z], since ^ = 1. 



87. To expand f(x + h) in a series of ascending powers 

 of h. 



Assume (Art. 85) that 



f(x + h) = A 9 + A 1 h + AJ l ? + AJi 3 + ......... (1), 



where A , A I} A v ..., do not contain h. 

 Then 



df(x + fy = dA h dA v fe , dA % ^ dA a { _ ^ ^Q\ 

 dx dx dx dx dx 



and *f (a h} = A, + 2A z h + 3AJS + ..................... (3). 



By Art. 86, the series (2) and (3) must be equal. Hence, 

 equating the coefficients of like powers of h, we have 



dx ' 



A = ldA * = l d3A 

 3 dx 1.2.3 dx a ' 



And by putting h = in (1), we find 



