Integration of O() terms of the Taylor series

[nawidgc]

Hello,

I have two functions say f1(β) and f2(β) as follows:

f1(β)=1/(aδ^2) + 1/(bδ) + O(1) ... (1)

and

f2(β)= c+dδ+O(δ^2) ... (2)



where δ = β-η and a,b,c,d and η are constants. Eq. (1) and (2) are the Taylor series expansions of f1(β) and f2(β) about η respectively. I need to integrate f1(β) and f2(β) with respect to β (-1,1). Integration is straight forward for all the terms except O(1) and O(δ^2) in (1) and (2) respectively. How do I proceed here to integrate the O() terms? If anyone can guide me on this it will be extremely helpful. Many thanks for the help.
Regards,
N.

 

[fresh_42]

The basic problem is, that ##O(\cdot)## is unspecified, i.e. we do not know its exact value. I do not see how a definite integral would make sense here, as any value can result from it. For a generic behavior we have:

##O(1) = C_1## is simply a constant, so there is no problem with it.
##O(\delta^2)=O(\beta^2) \leq C_2\beta^2## yields only an upper bound, so integration will result in something less than ##C_3\cdot \beta^3##