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Coercion into series


Any time that an object like O[x] appears in a sum of terms,
Mathematica will convert the whole sum into a power series.

E.g.

a x + Exp[x] + O[x]^3
SeriesData[x, 0, {1, 1 + a, 1/2}, 0, 3, 1]

Cos[z] + O[z, Pi/2]^4
SeriesData[z, Pi/2, {-1, 0, 1/6}, 1, 4, 1]

and so on...

Consider the following equation

eq=(y + b)*(y + c) == a^4 + y^2 + 3;

Obviously

Solve[eq, {b, c}]
Solve::svars: Equations may not give solutions for all "solve"
variables.
{{b -> (3 + a^4 - c*y)/(c + y)}}

However how the appearance of O[y]^3 below "makes" Solve give solution
for {b,c}

Solve[(y + b)*(y + c) == (a^4 + y^2 + 3) + O[y]^3, {b, c}]
{{b -> -Sqrt[-3 - a^4], c -> Sqrt[-3 - a^4]}, {b -> Sqrt[-3 - a^4], c
-> -Sqrt[-3 - a^4]}}

Thanks a lot.

Dimitris


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