Hei.
Jeg har bare et kort spørsmål relatert til en del av en oppgavetekst. Teksten lyder:
Let C denote the positively oriented circle [tex] |z-2 |=1[/tex] on the Riemann surface for [tex]z^{1/2}[/tex], where the upper half of that circle lies on the sheet [tex]R_0[/tex] and the lower half on [tex]R_1[/tex]. Note that for each point [tex]z[/tex] on C, one can write
[tex]z^{1/2} = \sqrt {r}e^{i\Theta/2}[/tex]
where [tex]4\pi - \frac{\pi}{2} < \Theta < 4\pi + \frac{\pi}{2}[/tex]
Mitt spørsmål: Vil ikke denne definisjonsmengden for [tex]\Theta[/tex] kun definere høyre halvdel av den gitte sirkelen? Må ikke [tex]\Theta[/tex] være definert på en gren med intervall på [tex]2\pi[/tex]?
Setter stor pris på om noen kan forklare dette for meg!
Riemann surface
Moderators: Vektormannen, espen180, Aleks855, Solar Plexsus, Gustav, Nebuchadnezzar, Janhaa
Hei.
Oppgaven i sin helhet lyder som følger:
Let C denote the positively oriented circle [tex] |z-2 |=1[/tex] on the Riemann surface for [tex]z^{1/2}[/tex], where the upper half of that circle lies on the sheet [tex]R_0[/tex] and the lower half on [tex]R_1[/tex]. Note that for each point [tex]z[/tex] on C, one can write
[tex]z^{1/2} = \sqrt {r}e^{i\Theta/2}[/tex]
where [tex]4\pi - \frac{\pi}{2} < \Theta < 4\pi + \frac{\pi}{2}[/tex]
State why it follows that
[tex]\int_C z^{1/2} \, \mathrm{d}z = 0[/tex]
Generalize this result to fit the case of other simple closed curves that cross from one sheet to another without enclosing the branch points. Generalize to other functions, thus extending the Cauchy-Goursat theorem to integrals of multiple-valued functions.
Oppgaven i sin helhet lyder som følger:
Let C denote the positively oriented circle [tex] |z-2 |=1[/tex] on the Riemann surface for [tex]z^{1/2}[/tex], where the upper half of that circle lies on the sheet [tex]R_0[/tex] and the lower half on [tex]R_1[/tex]. Note that for each point [tex]z[/tex] on C, one can write
[tex]z^{1/2} = \sqrt {r}e^{i\Theta/2}[/tex]
where [tex]4\pi - \frac{\pi}{2} < \Theta < 4\pi + \frac{\pi}{2}[/tex]
State why it follows that
[tex]\int_C z^{1/2} \, \mathrm{d}z = 0[/tex]
Generalize this result to fit the case of other simple closed curves that cross from one sheet to another without enclosing the branch points. Generalize to other functions, thus extending the Cauchy-Goursat theorem to integrals of multiple-valued functions.