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enter image description hereHow did we go from the expression to the second. From the first expression I can write $f(\varphi(w))=\varphi(w^2)$ but I cant proceed further.

$\varphi^{-1} f(\varphi(w)))=w^2$ or $f(z)=\varphi(g(\varphi^{-1}(z)))$, where $g(w)=w^2$.

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    A little clarification needs here. – 2017-02-26
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    Its a pretty long question. i dont know how to type it. – 2017-02-26
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    done. now the first image is basically the question. – 2017-02-26
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    You have found it! But it's not related to conjugate w.? – 2017-02-26
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    yes mate.Just realised .stupid of me.Thanks though – 2017-02-26
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    I think the question is wrong. The question should be finding a conjugate phi(w) of f=(z^2)−2 – 2017-02-26
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    To be what? analytic? Harmonic? – 2017-02-26
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    I dont completely understand what you are saying.If you want to know this stuff go to week 2 lecture four of complex analysis at coursera.sorry. – 2017-02-26

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