I have added the remark to show what the question is referring to! Thank you!

Remark 1 Recall the proof of uniqueness of analytic solutions. a, n:=() = $1,0) = { coat". ar)". 2 2 C10+ (1) ( PEN Equating the constant" coefficients on both sides, with n= 1 and p = 0) = 4, gives 1.21 = (0,0 = P(C0,0), (2) a = 1.60,0 = Q(00,0) (3) For the first power, I', either p= 1 and q = 0, or p = 0 and q = 1, with ao = 0, as well as a1 = 000, 2-42 = C1,0 +0,121 = 10 +0,100,0 = 2.2(0,0,0,1,01,0). 21.0 +0.120.0 -C0,1000 = 02 (0,0,0,1,C10). (6) For r?, either p = 2 and q = 0 with C2,0-12. 1, or p = 1 and q = 1 with C1,1.1.21.th, or p = 0 and q = 2 with C0,2-40. (a.s)? or p=0 and q =1 with 0,1-40-22.12, with ag = 0, as well as a 1 = C1,0 and a2 10+0.100 3-43 = 22,0 +1,141 +0,2. (22.40 + 40-42 +41-91) +0,1-22 = C2,0 +1,100,0 +0.2. (22-0+0.42 +0.0-00,0) +00.142 (8) 41,0 +0.1.0.0 = (2,0 +01,100,0 +0,2-00,000,0 +0,1 2 = 3.0(0,0,0,1,1,0,0,2,C1,1,C2,0). (10) By induction, there are unique polynomials Q.., in the variables such that 4n = Qu(Cpa)p+osa) (11) where the coefficients of (n!), are non-negative integers, which do not depend on the coefficients of f. Cp Problem 3 Determine any two radii Ra >0 and R, >0 such that the function Y defined by Y(0:1) r. 1- 1+2 M 1-a In 1-5) (13) is holomorphic for all a and such that al 0 and r >0. Remark 1 Recall the proof of uniqueness of analytic solutions. a, n:=() = $1,0) = { coat". ar)". 2 2 C10+ (1) ( PEN Equating the constant" coefficients on both sides, with n= 1 and p = 0) = 4, gives 1.21 = (0,0 = P(C0,0), (2) a = 1.60,0 = Q(00,0) (3) For the first power, I', either p= 1 and q = 0, or p = 0 and q = 1, with ao = 0, as well as a1 = 000, 2-42 = C1,0 +0,121 = 10 +0,100,0 = 2.2(0,0,0,1,01,0). 21.0 +0.120.0 -C0,1000 = 02 (0,0,0,1,C10). (6) For r?, either p = 2 and q = 0 with C2,0-12. 1, or p = 1 and q = 1 with C1,1.1.21.th, or p = 0 and q = 2 with C0,2-40. (a.s)? or p=0 and q =1 with 0,1-40-22.12, with ag = 0, as well as a 1 = C1,0 and a2 10+0.100 3-43 = 22,0 +1,141 +0,2. (22.40 + 40-42 +41-91) +0,1-22 = C2,0 +1,100,0 +0.2. (22-0+0.42 +0.0-00,0) +00.142 (8) 41,0 +0.1.0.0 = (2,0 +01,100,0 +0,2-00,000,0 +0,1 2 = 3.0(0,0,0,1,1,0,0,2,C1,1,C2,0). (10) By induction, there are unique polynomials Q.., in the variables such that 4n = Qu(Cpa)p+osa) (11) where the coefficients of (n!), are non-negative integers, which do not depend on the coefficients of f. Cp Problem 3 Determine any two radii Ra >0 and R, >0 such that the function Y defined by Y(0:1) r. 1- 1+2 M 1-a In 1-5) (13) is holomorphic for all a and such that al 0 and r >0