# Question:Approximating a function near a point

## Question:Approximating a function near a point

Maple 2019

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 /

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