Question regarding harmonic functions having the MVP via proof in Gamelin complex analysis

Question

The question I have is regarding the following part in Gamelin's Complex Analysis. This is regarding showing that harmonic functions have the MVP (page 85-86). The step in question is the last equality in the equation 4.2. I am not seeing the transition between the previous step to the final step there, hopefully someone can enlighten me! equation 4.2

Answer 1

I believe here, the notation is the cause of confusion.

Note that, $$ \dfrac{\partial u}{\partial r}(z_0 + re^{i\theta}) $$ does NOT mean $$\dfrac{\partial u}{\partial r} \times (z_0 + re^{i\theta}) $$ but rather, $$\dfrac{\partial}{\partial r}u(z_0 + re^{i\theta}) .$$ So, $(z_0 + re^{i\theta})$ is the argument for the function $u$. However, $u$ is a function of $z$ where $z$ is defined as $x + iy$. In the proof, they have parameterised $x(\theta) = x_0+r\cos\theta$ and $y(\theta) = y_0 + r\sin\theta$. As such, \begin{align*} z &= x+iy \\ &= x_0+r\cos\theta+iy_0+ ir\sin\theta \\ &= (x_0+iy_0) + r(\cos\theta+i\sin\theta) \\ &= z_0+re^{i\theta}. \end{align*} So in fact, $$ u(z_0 + re^{i\theta}) = u(z) = u $$ and so $$ \dfrac{\partial u}{\partial r}(z_0 + re^{i\theta}) = \dfrac{\partial u}{\partial r}(z) = \dfrac{\partial u}{\partial r}. $$By the multivariable chain rule, $$ \dfrac{\partial u}{\partial r} = \dfrac{\partial u}{\partial x}\dfrac{\partial x}{\partial r} + \dfrac{\partial u}{\partial y}\dfrac{\partial y}{\partial r} $$ where $$\dfrac{\partial x}{\partial r} = \cos\theta \text{ and } \dfrac{\partial y}{\partial r} = \sin\theta. $$ Thus, $$ \dfrac{\partial u}{\partial r} =\dfrac{\partial u}{\partial x}\cos\theta+ \dfrac{\partial u}{\partial y}\sin\theta. $$ So wrapping it all together, $$ r\int_0^{2\pi}\dfrac{\partial u}{\partial r}(z_0 + re^{i\theta})\,d\theta = r\int_0^{2\pi}\dfrac{\partial u}{\partial r} \,d\theta=r\int_0^{2\pi}\left[\dfrac{\partial u}{\partial x}\cos\theta+ \dfrac{\partial u}{\partial y}\sin\theta\right ]\,d\theta. $$

As a side note, for further questions on MSE, make sure to include all necessary details in your question. For example, it is not obvious how $x$ and $y$ are parameterised, nor what function $u$ is dependent on, and so on. All these details are necessary to understand what has happened, but you have omitted it from the question.