Post by mondo on Sept 30, 2023 5:44:40 GMT
This subsection is very unclear to me, and I need some help to understand this content - here are the places I don't understand for now:
1. On page 560 in the second to last paragraph author says "By Schwarz's result, the temperature at $a$ is $T(a) = (K^*/2\pi)$" but then, on the top of the next page author says "$<<T>>_a - (\lambda/2\pi)$, where $\lambda$ is the angle subtended by $K$ at $a$". So from these two it would mean that $K^* = \lambda$ but how can that be? One is the arc length ($K^*$) while the other is an angle $\lambda$? If the angle is based at the origin then it could be true as we are on the unit circle but it is located at a point $a$ instead.
2. Author also says, $\lambda = \frac{1}{2}(K^* + K)$ this again seems to treat arc length and and an angle as the same quantity - what is it based on? Maybe there is some Euclid rule I missed here. But also there is contradiction, as I said in my questions #1 before author said $T(a) = (K^*/2\pi)$ now the claim is $T(a) = ((1/2(K^*+K))/2\pi)$ these two are not equal.
3. At the beginning of page 562, author gives an example from hyperbolic geometry here and writes an equation of some hyperbolic angle which is equal to $\lambda + (\cdot+ \odot)$ I again have no clue what angle is it?
4. Last paragraph of page 562 - "Reinterpreting (32) we now see that $d\theta^*$ is simply the hyperbolic angle subtended at $a$ by the element of $C$" so why is $d\theta^*$ the hyperbolic angle? I thought that $z^*$ and $\theta^*$ are values due to a mapping function $h(z)$ described at the bottom of page 556 and that has nothing to do with a hyperbolic view of the plane.
Thank you.
1. On page 560 in the second to last paragraph author says "By Schwarz's result, the temperature at $a$ is $T(a) = (K^*/2\pi)$" but then, on the top of the next page author says "$<<T>>_a - (\lambda/2\pi)$, where $\lambda$ is the angle subtended by $K$ at $a$". So from these two it would mean that $K^* = \lambda$ but how can that be? One is the arc length ($K^*$) while the other is an angle $\lambda$? If the angle is based at the origin then it could be true as we are on the unit circle but it is located at a point $a$ instead.
2. Author also says, $\lambda = \frac{1}{2}(K^* + K)$ this again seems to treat arc length and and an angle as the same quantity - what is it based on? Maybe there is some Euclid rule I missed here. But also there is contradiction, as I said in my questions #1 before author said $T(a) = (K^*/2\pi)$ now the claim is $T(a) = ((1/2(K^*+K))/2\pi)$ these two are not equal.
3. At the beginning of page 562, author gives an example from hyperbolic geometry here and writes an equation of some hyperbolic angle which is equal to $\lambda + (\cdot+ \odot)$ I again have no clue what angle is it?
4. Last paragraph of page 562 - "Reinterpreting (32) we now see that $d\theta^*$ is simply the hyperbolic angle subtended at $a$ by the element of $C$" so why is $d\theta^*$ the hyperbolic angle? I thought that $z^*$ and $\theta^*$ are values due to a mapping function $h(z)$ described at the bottom of page 556 and that has nothing to do with a hyperbolic view of the plane.
Thank you.
