Post by Admin on Aug 11, 2016 15:50:05 GMT
Gary
From readings of Section II Spherical Geometry, and attempting exercise 5, I feel fairly confident that I have discovered a couple of errors on page 286. The second error results from the first. The first error is in the fourth equation on the page, which I think should be
$$|f'(z)|=\frac{\Lambda(z)}{\Lambda[f(z)]}$$ and then (17) which should be
$$|f'(z)|=\frac{1+|f(z)|^2}{1+|z|^2}.$$
The reason I think this is because from the third equation on the page we have
$$\frac{d\tilde{s}}{ds}=\frac{\Lambda(z)}{\Lambda[f(z)]}$$ and since $$\frac{d\tilde{s}}{ds}=\frac{|d\tilde{z}|}{|dz|}=|f'(z)|$$ we must have
$$|f'(z)|=\frac{\Lambda(z)}{\Lambda[f(z)]},$$ and the error in (17) follows from this.
This also ties in with the fact that $\Lambda$ is a real number.
What do you think?
Vasco
From readings of Section II Spherical Geometry, and attempting exercise 5, I feel fairly confident that I have discovered a couple of errors on page 286. The second error results from the first. The first error is in the fourth equation on the page, which I think should be
$$|f'(z)|=\frac{\Lambda(z)}{\Lambda[f(z)]}$$ and then (17) which should be
$$|f'(z)|=\frac{1+|f(z)|^2}{1+|z|^2}.$$
The reason I think this is because from the third equation on the page we have
$$\frac{d\tilde{s}}{ds}=\frac{\Lambda(z)}{\Lambda[f(z)]}$$ and since $$\frac{d\tilde{s}}{ds}=\frac{|d\tilde{z}|}{|dz|}=|f'(z)|$$ we must have
$$|f'(z)|=\frac{\Lambda(z)}{\Lambda[f(z)]},$$ and the error in (17) follows from this.
This also ties in with the fact that $\Lambda$ is a real number.
What do you think?
Vasco
