Increment of the argument along a curve

Let $f$ be the analytic function defined in the disk $\Delta = \left\{ z : \left|z-4\right| < 4\right\}$ so that $f(z) = z^{1/3}(z+1)^{1/2}$ in $\Delta$ and $f(x)$ is positive for $0 < x < 8$. An analytic function $g$ in $\Delta$ is obtained from $f$ by analytic continuation along the path starting and ending at $z = 4$ (see Fig. 5.2). Express $g$ in terms of $f$.

Trace how the argument of the quantity $x(x-1)$ changes as $x$ traces, counterclockwise, a closed curve enclosing the points $0$ and $1$.

Study how the arguments of the quantities $\sqrt{x}$, $\sqrt{x-1}$, and $\sqrt{x(x-1)}$ change as $x$ traces the same contour as in problem 2480 (a closed curve traversed counterclockwise and enclosing the points $0$ and $1$).

How does the function $u=(z-a)^{\alpha }(z-b)^{\beta }$ change, for real $\alpha$ and $\beta$, if $z$ traces a closed contour going counterclockwise around the points $a$ and $b$?

The same question for the function $u=(z-a)^{\alpha }\ln (z-a)$ and a contour going counterclockwise around the point $a$.

Study the change in $\arg \sqrt[n]{z-a}$ as the point $z$ traces a simple closed contour going around the point $z = a$ in the positive direction.

Study the change in the function $\sqrt{1-z^{2}}$ as the point $z$ traces a simple closed contour of one of three kinds:

Find the increment of the argument of the function $z(\cdot ) \in C^{1}[0,1]$ on the segment $[0,1]$, given in the form $z(t) = x(t)+i y(t)$, where

Find the increment of the argument of the function $z(\cdot ) \in C^{1}[0,1]$ on the segment $[0,1]$, given in the form $z(t) = x(t)+i y(t)$, where

Compute the increment of the argument of the function $z^{2}+1$ along the oriented curve $\gamma$, given in the form $z(t) = x(t)+i y(t)$, where

Compute the increment of the argument of the function $\frac{1}{z^{2}+2 z}$ along the oriented curve $\gamma$, given in the form $z(t) = x(t)+i y(t)$, where

Compute the increment of the argument of the function $\frac{z^{2}+1}{z^{2}-4}$ along the oriented curve $\gamma$, given in the form $z(t) = x(t)+i y(t)$, where

Compute the increment of the argument of the function $\frac{\operatorname {tg} z}{z}$ along the oriented closed curve given by the equation

when it is traversed once counterclockwise.

Compute the increment of the argument of the function $\frac{\mathrm{ch}^{2} z}{z}$ along the oriented closed curve given by the equation $\left|x\right|+\left|y\right| = 3$, when it is traversed once clockwise.

Prove the equality

for any simple piecewise-smooth closed curve $\gamma$ such that the domain bounded by the curve $\gamma$ contains the segment $[0,1]$.