Shahada ya kwanza (juu)AnalysisImepangwa
Complex analysis
Holomorphic functions and the Cauchy–Riemann equations, contour integration and Cauchy's theorem, power series, Liouville's theorem, the residue theorem and conformal maps.
math.analysis.complex-analysis
Somo la mada hii bado halijaandikwa. Muhtasari ulio hapa chini ni sehemu ya ramani ya mtaala na haujakaguliwa.
Inajengwa juu ya
Masharti ya awali ya lazima 39 (hadi chini kabisa)
Utajifunza
- Check holomorphy using the Cauchy–Riemann equations.
- Prove Cauchy's integral formula and deduce that holomorphic functions are analytic.
- Evaluate real integrals by residues, and prove the fundamental theorem of algebra from Liouville's theorem.
Fasili
- f is holomorphic on an open set U ⊆ ℂ if it is complex-differentiable at every point of U.
- The residue of f at an isolated singularity a is the coefficient of (z − a)⁻¹ in its Laurent expansion about a.
Nadharia
- Cauchy's integral formula.
- Holomorphic functions are infinitely differentiable and are locally equal to their Taylor series.
- Liouville: every bounded entire function is constant.
- Fundamental theorem of algebra: every non-constant complex polynomial has a complex root.
Mawazo potofu ya kawaida
- Assuming every real-differentiable function of two variables is holomorphic; z ↦ z̄ is not.
- Thinking a function holomorphic except at one point always extends holomorphically to that point; 1/z does not.
Ugunduzi wa hiari
- Algebraic geometry Compact Riemann surfaces correspond to smooth projective complex algebraic curves.
- ← Calculus of one variable
Njia za kujifunza
Usomaji zaidi
- Lars V. Ahlfors, Complex Analysis (1953)
- Elias M. Stein and Rami Shakarchi, Complex Analysis (2003)