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I am taking a first course on complex analysis and I am wondering whether we can differentiate complex functions as if they were real.

Most of the functions I have come across i.e. $\cos z, \sin z, \cosh z$ etc. follow the same rules of derivation as their real counterparts.

Are there certain conditions that allow me to make this assumption?

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Yes, the differentiation formulas for analytic functions are the same as their real versions. That is, if $f$ and $g$ are analytic functions on some domain $D$ whose intersection $D \cap \mathbb R$ with the real line contains an interval, and $f'(x) = g(x)$ for $x \in D \cap \mathbb R$, then by analytic continuation $f'(z) = g(z)$ for all $z \in D$.

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If you actually have a function only containing $z$ and not something like $f(u+iv)=u^2+iv^2$, you can apply all rules that work in the reals.

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Complex functions that can be differentiated as if $z$ is real must satisfy Cauchy-Riemann equations. That way they are said to be complex differentiable.

http://mathworld.wolfram.com/Cauchy-RiemannEquations.html http://mathworld.wolfram.com/ComplexDifferentiable.html

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