I realise this is probably a really obvious question...

I have this function:

(2^(1/3)*Pi*AiryAi(x))/(b*(a*AiryAi(x) + b*AiryBi(x)))

And I want to approximate it near to the root of the equation

a*AiryAi(x) + b*AiryBi(x)=0 (say the solution is x=x0)

The first term - I know - is a term for 1/(x-x0) for which the coefficient is

AiryAi(RootOf(AiryAi(_Z)*a + b*AiryBi(_Z)))*2^(1/3)*Pi/(b*a*AiryAi(1, RootOf(AiryAi(_Z)*a + b*AiryBi(_Z))) + b^2*AiryBi(1, RootOf(AiryAi(_Z)*a + b*AiryBi(_Z)))).

However the next term, which should be a constant, comes out as a really large term which I can't make sense of (I've included it at the bottom of this post but it is very large).

Am I doing something wrong? Because I feel that this should be easier than I'm making it.

The determined expansion was:

(1/3) 2

-3 2 Pi RootOf(AiryAi(_Z) a + b AiryBi(_Z))

3

AiryBi(RootOf(AiryAi(_Z) a + b AiryBi(_Z))) AiryAi(RootOf(

3 (1/3)

AiryAi(_Z) a + b AiryBi(_Z))) b - 9 2 Pi

2

RootOf(AiryAi(_Z) a + b AiryBi(_Z))

2

AiryBi(RootOf(AiryAi(_Z) a + b AiryBi(_Z)))

2 2 (1/3)

AiryAi(RootOf(AiryAi(_Z) a + b AiryBi(_Z))) a b - 9 2 Pi

2

RootOf(AiryAi(_Z) a + b AiryBi(_Z)) AiryBi(RootOf(AiryAi(_Z) a

3

+ b AiryBi(_Z))) AiryAi(RootOf(AiryAi(_Z) a + b AiryBi(_Z)))

2 (1/3) 2

a b - 3 2 Pi RootOf(AiryAi(_Z) a + b AiryBi(_Z))

4 3 (1/3)

AiryAi(RootOf(AiryAi(_Z) a + b AiryBi(_Z))) a - 4 2 Pi

RootOf(AiryAi(_Z) a + b AiryBi(_Z)) AiryBi(RootOf(AiryAi(_Z) a

+ b AiryBi(_Z)))

2

AiryBi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z))) AiryAi(RootOf(

3 (1/3)

AiryAi(_Z) a + b AiryBi(_Z))) b - 8 2 Pi RootOf(AiryAi(_Z) a

+ b AiryBi(_Z)) AiryBi(RootOf(AiryAi(_Z) a + b AiryBi(_Z)))

AiryBi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z))) AiryAi(1,

RootOf(AiryAi(_Z) a + b AiryBi(_Z))) AiryAi(RootOf(AiryAi(_Z) a

2 (1/3)

+ b AiryBi(_Z))) a b - 4 2 Pi RootOf(AiryAi(_Z) a

+ b AiryBi(_Z)) AiryBi(RootOf(AiryAi(_Z) a + b AiryBi(_Z)))

2

AiryAi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z))) AiryAi(RootOf(

2 (1/3)

AiryAi(_Z) a + b AiryBi(_Z))) a b - 4 2 Pi RootOf(AiryAi(

_Z) a + b AiryBi(_Z))

2

AiryBi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))

2 2 (1/3)

AiryAi(RootOf(AiryAi(_Z) a + b AiryBi(_Z))) a b - 8 2 Pi

RootOf(AiryAi(_Z) a + b AiryBi(_Z)) AiryBi(1,

RootOf(AiryAi(_Z) a + b AiryBi(_Z))) AiryAi(1,

RootOf(AiryAi(_Z) a + b AiryBi(_Z)))

2 2 (1/3)

AiryAi(RootOf(AiryAi(_Z) a + b AiryBi(_Z))) a b - 4 2 Pi

RootOf(AiryAi(_Z) a + b AiryBi(_Z))

2

AiryAi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))

2 3 (1/3)

AiryAi(RootOf(AiryAi(_Z) a + b AiryBi(_Z))) a + 2 2 Pi

2

AiryBi(RootOf(AiryAi(_Z) a + b AiryBi(_Z))) AiryBi(1,

RootOf(AiryAi(_Z) a + b AiryBi(_Z))) AiryAi(RootOf(AiryAi(_Z) a

3 (1/3)

+ b AiryBi(_Z))) b + 2 2 Pi

2

AiryBi(RootOf(AiryAi(_Z) a + b AiryBi(_Z))) AiryAi(1,

RootOf(AiryAi(_Z) a + b AiryBi(_Z))) AiryAi(RootOf(AiryAi(_Z) a

2 (1/3)

+ b AiryBi(_Z))) a b + 4 2 Pi AiryBi(RootOf(AiryAi(_Z) a

+ b AiryBi(_Z))) AiryBi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))

2 2 (1/3)

AiryAi(RootOf(AiryAi(_Z) a + b AiryBi(_Z))) a b + 4 2 Pi

AiryBi(RootOf(AiryAi(_Z) a + b AiryBi(_Z))) AiryAi(1,

RootOf(AiryAi(_Z) a + b AiryBi(_Z)))

2 2 (1/3)

AiryAi(RootOf(AiryAi(_Z) a + b AiryBi(_Z))) a b + 12 2 Pi

3

AiryBi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z))) AiryAi(1,

3 (1/3)

RootOf(AiryAi(_Z) a + b AiryBi(_Z))) b + 36 2 Pi

2

AiryBi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))

2 2

AiryAi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z))) a b + 36

(1/3)

2 Pi AiryBi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))

3 2 (1/3)

AiryAi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z))) a b + 2 2 Pi

AiryBi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))

3 2

AiryAi(RootOf(AiryAi(_Z) a + b AiryBi(_Z))) a b

(1/3) 4

+ 12 2 Pi AiryAi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))

3 (1/3)

a + 2 2 Pi AiryAi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))

3 3\//

AiryAi(RootOf(AiryAi(_Z) a + b AiryBi(_Z))) a / \12

4 4

AiryAi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z))) a b + 48

3

AiryAi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z))) AiryBi(1,

3 2

RootOf(AiryAi(_Z) a + b AiryBi(_Z))) a b + 72

2

AiryAi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))

2 2 3

AiryBi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z))) a b + 48

AiryAi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z)))

3 4

AiryBi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z))) a b

4 5\

+ 12 AiryBi(1, RootOf(AiryAi(_Z) a + b AiryBi(_Z))) b /