1 86 MATHEMATICAL FORMULAE AND ELLIPTIC FUNCTIONS 



where 



_ _ , a. x a(a+ i) x 2 a(a + i)(a+ 2) x* 



The series is absolutely and uniformly convergent for all real and complex values 

 of a, y, x, except when 7 is a negative integer or zero. 



When 7 is a positive integer the complete solution of the differential 

 equation is: 



y = d + c 2 log x M(a, 7, x) + c 2 - - i 



a(a + i) x? /_i i ^ _ * _ _ A 

 7(7 + 1) 2!\a a+i 7 7+1 27 



q(cg + i)(a + 2) ^ / _i i i _i _ i i _ _ i _ i\ 



7(7 + *)(7 + 2) 3! \ a + i a+2 7 7 + 1 7 + 2" "2" 37 



8.601 For large values of x the following asymptotic expansion may be used 

 M(<x, 7, *) 



2 



i 2! 



8.61 



1. M(a, 7, ) = e?M(y-a, 7, -) 



2. x l ~^M(a -7 + 1, 2 - 7, x) = e x x l -t M(i -a, 2-7, -x). 



3. ^ M(a + i, 7 + i, *) = M(a + i, 7, x) - M(a, 7, *). 



4. aM (a + i, 7 + i, #) = (a - 7)Jlf (a, 7 + 1,0:) + 7M(a, 7, x). 



5. (a + *)Jlf (a + i, 7 + i, *) = (a - y)M(a, y + i, x) + yM(a + i, 7, *) 



6. ayM(a + i, 7, x) = y(a + )M (a, 7, *) - ^(7 - a)M(a, 7 + 1, ^). 



7. aM(a + i, 7, *) = (x + 20: - 7)M(a, 7, *) + (7 - a)^(a - i, 7, *)- 



8 - 7 ~ ^^^(a, 7 + i, a) = (x + 7 - i)Af(a, 7, a) + (i - 7)^(, 7 - i, 



8.62 



I- JW(a, 7, *) = AT (a +i, 7 + 1, x). 



. (i - a) I M(a, 7, 



*) ^ = (i - 7)M(a - i, 7 - i, *) + (7 - i). 



