The Gamma function and conic sections

The Gamma function

For $z\in\CC$ with $\operatorname{Re}z>0$, the gamma function is \[ \Gamma(z)\coloneqq\int_0^\infty t^{z-1}e^{-t}\dt, \] where $t^{z-1}\coloneqq e^{(z-1)\log t}$ for $t>0$, with $\log$ the real natural logarithm.

The poles, residues, product expansion, and reflection formula of $\Gamma$ are on Special functions.

Uniformization theorem (Tag T-ZAEWG)

Every connected Riemann surface is a quotient of one simply connected model: $\CP^1$, $\CC$, or $\DD$.Suggest a slogan

Let $S$ be a connected Riemann surface with universal cover $\widetilde S\to S$. Then $\widetilde S$ is a simply connected Riemann surface, and its group $G$ of deck transformations acts on $\widetilde S$ freely and properly discontinuously by biholomorphisms, with $S$ biholomorphic to $\widetilde S/G$. Moreover, $\widetilde S$ is biholomorphic to exactly one of $\CP^1$, $\CC$, and $\DD$.

Conic sections

(Standard forms of conic sections)

Circle: $x^2 + y^2 = r^2$.
Ellipse: $\qty{\frac x a}^2 + \qty{\frac y b}^2 = 1$.
Hyperbola: $\qty{\frac x a}^2 - \qty{\frac y b}^2 = 1$; the rectangular hyperbola $xy = \frac{c^2}{2}$.
Parabola: $y^2 = 4ax$.

(Classification by the discriminant)

For a nondegenerate real conic $Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0$ with at least one real point and discriminant $\Delta \coloneqq B^2 - 4AC$:

$\Delta < 0$ if and only if the conic is an ellipse, and it is a circle if and only if moreover $A=C$ and $B=0$;
$\Delta = 0$ if and only if the conic is a parabola;
$\Delta > 0$ if and only if the conic is a hyperbola.

(Completing the square)

With $s \coloneqq \frac{b}{2}$, \[ \begin{aligned} x^2 - bx &= (x - s)^2 - s^2, \\ x^2 + bx &= (x + s)^2 - s^2. \end{aligned} \]

The interior angles of a convex $n$-gon sum to $(n-2)\pi$, so each interior angle of a regular $n$-gon is $\frac{n-2}{n}\pi$.

No continuous square root on $\CC$ (Tag E-6H7HD)

Show that there is no continuous square root function defined on all of $\CC$.

Solution

Suppose $f(z)^2 = z$. Then $f$ is a section to the covering map \[ p: \unitsof{\CC} &\to \unitsof{\CC} \\ z & \mapsto z^2 ,\] so $p\circ f = \id$. Using $\pi_1(\unitsof{\CC}) = \ZZ$, the induced maps are $p_*(1) = 2$ and $f_*(1) = n$ for some $n\in \ZZ$. But then $p_* \circ f_*$ is multiplication by $2n$, contradicting $p_* \circ f_* = \id$ by functoriality.

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Appendices