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Is it mathematically correct to say that $$\frac{d}{dt}(a+bi) = if(t)(a+bi)$$ implies that $$(a+bi) = Ne^{ig(t)}$$ where $N$ is a complex constant and $\frac{d}{dt}{g}=\frac{f}{N}$?

I'm just not sure whether the reasoning is sound because of the complex nature of the functions

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    Actually, $z'(t)=if(t)z(t)$ is equivalent to $z(t)=Ce^{ig(t)}$ where $C$ is any complex constant and $g'(t)=f(t)$, not $g'(t)=f(t)/C$ as you wrote. – 2017-01-16
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    Oh, right cause $N=e^{iK}$ for some $K$ and then you have $e^{i(g(t) + K)}$. – 2017-01-16
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    @JanEerland Offtopic. Everybody understood that $a$ and $b$ are functions of $t$. – 2017-01-16

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