Conformal change of variables in derivatives and the Laplacian

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Problem 3.55.1 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Problem 3.55 "[NO TITLE]"

Assume that $f(z)$ is analytic, use the notation

w = u+iv = f(z) = f(x+iy)

as above and use subscripts to denote partial derivatives in the usual way. Verify that

u_{x}^{2} + v_{x}^{2} = u_{y}^{2} + v_{y}^{2} = u_{x}^{2} + u_{y}^{2} = v_{x}^{2} + v_{y}^{2} = u_{x}v_{y} - u_{y}v_{x} = \left|\frac{dw}{dz}\right|^{2}.

Problem 3.55.3 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Problem 3.55 "[NO TITLE]"

(continued). Let $\varphi (x,y)$ and $\psi (x,y)$ denote functions of the two real variables $x$ and $y$ having continuous derivatives of first order; they can also be considered as functions of $u$ and $v$ where $f'(z) \neq 0$ . Verify that

\varphi _{x}\psi _{x} + \varphi _{y}\psi _{y} = (\varphi _{u}\psi _{u} + \varphi _{v}\psi _{v})\left|\frac{dw}{dz}\right|^{2}.

Problem 3.55.4 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Problem 3.55 "[NO TITLE]"

(continued). If $\varphi (x,y)$ has continuous derivatives of the second order, also

\varphi _{xx} + \varphi _{yy} = (\varphi _{uu}+\varphi _{vv})\left|\frac{dw}{dz}\right|^{2}.

Problem 3.55.5 from Pólya G., Szegő G. "Problems and Theorems in Analysis" [PóSz98]More problems in Problem 3.55 "[NO TITLE]"

(continued). Assume that $a, b, c$ and $d$ are real constants,

ad - bc = 1,

and consider

w = \frac{az+b}{cz+d}.

Then

\frac{\varphi _{u}^{2}+\varphi _{v}^{2}}{\varphi _{x}^{2}+\varphi _{y}^{2}} = \frac{\varphi _{uu}+\varphi _{vv}}{\varphi _{xx}+\varphi _{yy}} = \frac{y^{2}}{v^{2}}.

References

[PóSz98] Pólya G., Szegő G. Problems and Theorems in Analysis. Springer, 1998.