The Cauchy-Riemann Differential Equations

Are the functions

analytic?

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

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

(continued). Prove that

(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

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

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

and consider

Then

Find the analytic function of $z$ that vanishes for $z = 0$ and has the real part

We denote by $a$ and $b$, $a < b$, two fixed real numbers, by $z$ a variable point in the half-plane $\mathrm{Im}\left(z\right) > 0$ and by $\omega$ the variable angle under which the interval $[a,b]$ is seen from the point $z$. If possible find an analytic function the real part of which is $\omega$.

Show that for any analytic function $f(z) = f(x+iy)$

Show that for any analytic function of $z = x+iy$