考试大纲（大学组一）

 

考试内容主要包微积分内容，总范围不超过CMC（非数学类）与考研数学一，此处不详细展开。

 

考试大纲（大学组二）

 

Real analysis

1. Review of metric spaces. Open and closed sets. Convergence. Complete spaces and compact spaces. Continuous and uniformly continuous functions.
2. Construction of a measure on a \( \sigma \)-algebra of subsets. Outer measure. Measurable sets and sets of measure zero.
3. The Lebesgue measure on \( \mathbb R^n \) . Borel sets and sets of measure zero. The \(\sigma\)-algebra of Lebesgue measurable sets. Approximation by open and closed sets. Measurable functions. Egoroff’s theorem, Lusin’s theorem.
4. Integration. Construction of the Lebesgue integral. Basic properties. Fatou’s lemma. Monotone convergence theorem. Dominated convergence theorem. Product measures and Fubini’s theorem on \( \mathbb R^n \) .
5. The Radon- Nikodim theorem. The Lebesgue differentiation theorem. Functions of bounded variation on the real line. Absolutely continuous functions.
6. \( L^p \) spaces. Measure spaces and measurable functions. The spaces \( L^1 \) and \(L^2\) , including completeness. The inner product in \( L^2 \) . Convolutions and their continuity and smoothing properties.
7. The Fourier transform on \(L^2\) (\( \mathbb R^n ) \) . Definition and basic properties. Inversion formula and Plancherel’s theorem. Applications to PDE’s with constant coefficients.

 

Complex analysis

1. Basics. Geometric description of complex numbers. The complex plane and the Riemann sphere. Conformal mappings. Linear transformations as conformal maps. Representation by complex numbers.
2. Holomorphic functions. Definition. Exponential and trigonometric functions. Conformality and the Cauchy-Riemann equations. Relation to harmonic functions. Power series: uniform convergence, Weierstrass’ Mtest, continuity, integrability, differentiability. Integration along curves, primitives.
3. Cauchy’s Theorem and Applications. Goursat’s theorem. Cauchy’s theorem on a disk. Evaluation of integrals. Morera’s theorem. Cauchy integral formulas. Cauchy estimates and Liouville’s theorem. Fundamental theorem of algebra. Isolated zeros and analytic continuation. Sequences of holomorphic functions. Schwarz reflection principle.
4. Meromorphic functions. Zeros and poles. Laurent series. The residue formula for some domains. Jordan and “small arc” lemmas. Computation of integrals by residue calculus. Riemann’s theorem on removable singularities. Essential singularities and Casorati-Weierstrass theorem. The argument principle and applications (Rouché’s theorem, open mapping theorem, maximum modulus theorem).
5. Plane topology. Simply and multiply connected domains. Jordan curve theorem (without proof). Homotopies and the winding number. General form of Cauchy’s theorem. Roots and logarithm (including branches and cuts). Additional examples of computation of integrals using residues.
6. Conformal maps. Elementary conformal maps. Schwarz lemma and automorphisms of the disk and upperhalf plane. Fractional linear transformations (cross ratio, behavior of lines and circles). Normal familes. Montel’s theorem and the Riemann mapping theorem.

 

Functional analysis

1.Normed spaces. Banach spaces. Linear operators. Examples.
2. Spaces of bounded linear operators. The uniform boundedness principle and the open mapping theorem.
3. Bounded linear functionals. Dual spaces. The Hahn-Banach extension theorem. Separation of convex sets.
4. Spaces of continuous functions. Ascoli’s theorem, Stone-Weierstrass’ theorem. The space \( C^{0,\gamma } \) of Holder continuous functions and the space \(C^k \) of \(k \)-times differentiable functions.
5. Hilbert spaces. Perpendicular projections. Orthonormal bases. Selfadjoint operators.
6. Compact operators on a Hilbert space. Fredholm’s alternative. Spectrum and eigenfunctions of a compact, self-adjoint operator. Applications to Sturm-Liouville boundary value problems.

 

ODEs and PDEs

1.Existence and uniqueness theorems for solutions of ODE; explicit solutions of simple equations.
2.self-adjoint boundary value problems on finite intervals; critical points, phase space, stability analysis
3.First order partial differential equations, linear and quasi-linear PDE.Heat equation, Dirichlet problem, fundamental solutions. Green functions and existence of solutions of Dirichlet problem, harmonic functions, maximal principle and applications, existence of solutions of Neumann’s problem