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So assume ƒ is differentiable at 0, as a function of two real variables from Ω to C. This is equivalent to the existence of two complex numbers and ( which are the partial derivatives of ƒ ) such that we have the linear approximation
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So and assume
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So and ƒ
To justify the interchange of the sum and the integral, one must notice that in the intersection of |( z − a )/( w − a )| ≤ r < 1 and some closed domain containing C, ƒ ( w )/( w − a ) is holomorphic and therefore bounded by some positive number M. So we have
So each critical point z < sub > 0 </ sub > of ƒ lies at the center of a disc B ( z < sub > 0 </ sub >, r ) containing no other critical point of ƒ in its closure.
So, formally, it follows that the Fourier series for the derivative is simply in multiplied by the Fourier series for ƒ.
So and is
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