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Potential phi as a function of x, y and z in the vicinity of the ring axis

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A thin circular ring of radius R lies in x-y plane and consists of two homogeneously opposite charged halves with charges q at x>0 and -q at x<0.

Find the potential phi as a function of x, y and z in the vicinity of the ring axis (x,y<<(R^2+z^2)/R) to linear order in x and y.

Answer is: phi= q*R*x/(pi^2*epsilonnot*(R^2+z^2)^3/2).

The question is - how to prove it.

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The expert examines the potential phi as a function of x, y and z in the vicinity of the ring axis.

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Problem:
A thin circular ring of radius R lies in x-y plane and consists of two homogeneously opposite charged halves with charges q at x>0 and -q at x<0.
Find the potential phi as a function of x, y and z in the vicinity of the ring axis( x,y<<(R^2+z^2)/R) to linear order in x and y.
Answer is given: phi= q*R*x/(pi^2*epsilonnot*(R^2+z^2)^3/2)

Explanation:
Electric potential at point r ⃗=(x,y,z) produced by any charge distribution ρ((r') ⃗) is ...

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