DEVELOPMENT OF FUNCTIONS 108 



<;;*. Development of Functions. If z^f(x+y) where x and y 

 are independent variables, 



fdz\ fdz\ 



Then (-5- =VT- 



\ay/x const M&r/v const 



for let 2 = f(w) where w = x + y 



dz dz dw 

 Keeping x constant -r- = -7- -y- 



</// aw '/// 



/da>\ 

 since (-7-) 



\dy /x const 



dz dz dw 

 Keeping y constant -r- = -7 -j- 



fi3P f I ~ i' fl^f* 



Us** | - - _ 



= -j- , since I -j ) =1 



UW \CuC / V const 



Thus f^r) = (^) 



\ay/x const \uX/y const 



The function s = /(a; + y) can be expressed as a series of terms 

 of descending powers of x, or ascending powers of y. 



Then z =f(x+y) = A + By + Cy 2 + Dy 3 + . . . 



where the coefficients A, B, C . . . are functions of x but are 

 independent of y. 



Then = B+ 2Cy+ 3Dy* + 



\dy/x const 



Afe\ dA , dB dC , , dD 



and (-r- =Tr+2/-T- + 2/j- + 2/^-+--- 



\da-Vyconst dx * dx dx dx 



Since these two expansions are equal, we can equate coefficients 

 of like powers of y. 



And B = t 



^ 

 <te~ dx* 2 



_ 



" " 



=_ 

 2 dec 3 = 8 



dD_ 1 



= d^~ |_3 d^ r 



dA r/ 2 d 2 A y 5 d?A 

 Therefore 2 T /(o: + 2/ ) = A + 2/ ^ + -^ _ + ^ + . . . , 



in which A can be easily expressed as a function of x by putting 

 y = 0, for then A =/(#). This expansion is known as Taylor's 

 Expansion. 



