A continuous argument computed along a spiralling contour

Given: The contour $\gamma (t)=(1+t)\exp (2\pi it)$ on $[0,1]$ and the point $p=0$.

A continuous logarithm of $\gamma -p$ along $\gamma$ is a continuous $\lambda$ with $\exp (\lambda (t))=\gamma (t)-p$ for every $t$, and its continuous argument is $\mathrm{Im}\lambda$ (Continuous logarithms and continuous arguments along a contour, The complex exponential by its power series).

For a complex contour $\gamma$ and $p\mathrm{\notin }{\gamma }^{\ast }$ there is a continuous logarithm of $\gamma -p$ along $\gamma$, and any two differ by a constant in $2\pi i\mathbb{Z}$ (Every contour missing a point admits a continuous logarithm, unique up to a constant in $2\pi i\mathbb{Z}$).

For a complex contour $\gamma :[a,b]\to \mathbb{C}$, a point $p\mathrm{\notin }{\gamma }^{\ast }$ and a continuous logarithm $\lambda$ of $\gamma -p$ along $\gamma$, ${\int }_{\gamma }dz\mathrm{/}(z-p)=\lambda (b)-\lambda (a)$ (The integral of $dz\mathrm{/}(z-p)$ along a contour is the increment of a continuous logarithm).

For $x>0$, $\log x$ is the unique real $y$ with $\exp y=x$ (The natural logarithm as the inverse of the exponential function); the natural logarithm is continuous and strictly increasing on $(0,\mathrm{\infty })$, and $\log 1=0$ (Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm).

A complex contour is a rectifiable path, and it is closed when its two endpoint values agree (Rectifiable complex contours, reversal, concatenation, closedness, and orientation); a continuous path differentiable with a continuous derivative on each piece of a partition is rectifiable (A continuous piecewise-$C^1$ path is rectifiable and its length is the sum of the speed integrals over its pieces).

$\exp (z+w)=\exp z\exp w$ ($\exp (z+w)=\exp z\exp w$, and the complex exponential extends the real exponential), and for real $x,y$, $\exp (x+iy)=e^x(\cos y+i\sin y)$ with $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$; in particular $\exp (2\pi i)=1$ ($\exp (x+iy)=e^x(\cos y+i\sin y)$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$, and $e^{i\pi }+1=0$, Quarter-turn values and shifts by pi/2 and pi).

For $z=a+bi$ with $a,b$ real, $\mathrm{Im}z=b$ (Real and imaginary parts, complex conjugation, and modulus).

$\cos$ and $\sin$ are differentiable with ${\cos }^{\mathrm{\prime }}=-\sin$ and ${\sin }^{\mathrm{\prime }}=\cos$ (The derivatives of sine and cosine are cosine and minus sine).