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| 3 | +title: Analysis General Exam Syllabus |
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| 127 | +</style> |
| 128 | + |
| 129 | +<article class="syllabus-container" role="main" aria-labelledby="syllabus-title"> |
| 130 | + <h1 id="syllabus-title">Analysis General Exam Syllabus</h1> |
| 131 | + <p class="revision-date">Revised March 2023</p> |
| 132 | + |
| 133 | + <section aria-labelledby="basics-section"> |
| 134 | + <h2 id="basics-section">Basics (common for complex and real parts)</h2> |
| 135 | + <ol class="topic-list"> |
| 136 | + <li>Elementary set operations, countable and uncountable sets.</li> |
| 137 | + <li>Open, closed, compact, and connected sets on the line and in Euclidean space.</li> |
| 138 | + <li>Completeness, infima and suprema, limit points, lim inf, lim sup.</li> |
| 139 | + <li>Bolzano-Weierstrass, Heine-Borel theorems.</li> |
| 140 | + <li>Continuous, uniformly continuous, differentiable functions.</li> |
| 141 | + <li>Extreme value, intermediate value, mean value theorems.</li> |
| 142 | + </ol> |
| 143 | + </section> |
| 144 | + |
| 145 | + <section aria-labelledby="complex-analysis-section"> |
| 146 | + <h2 id="complex-analysis-section">Complex Analysis</h2> |
| 147 | + <ol class="topic-list"> |
| 148 | + <li>Series of functions, power series, and power series of elementary functions, uniform convergence, Weierstrass <math><mi>M</mi></math> test. Formula for the radius of convergence of a power series.</li> |
| 149 | + <li>Analytic and harmonic functions, Cauchy-Riemann equations.</li> |
| 150 | + <li>Power series and Laurent series.</li> |
| 151 | + <li>Elementary conformal mappings, fractional linear mappings. The Cayley transform.</li> |
| 152 | + <li>Cauchy's integral theorem and Cauchy integral formula. Morera's theorem. Goursat's theorem.</li> |
| 153 | + <li>Power series representation is equivalent to complex differentiability, and the power series converges in any ball where the function is complex differentiable. Classification of singularities. Meromorphic functions. Casorati-Weierstrass (Sokhotski) theorem.</li> |
| 154 | + <li>Argument principle, open mapping theorem, maximum principle, Rouché's theorem, Schwarz's lemma, Liouville's theorem. Hurwitz' theorem.</li> |
| 155 | + <li>Cauchy's residue theorem, evaluation of definite integrals. Evaluation of infinite series via residue theory.</li> |
| 156 | + <li>Normal families. Montel's theorem. Riemann mapping theorem.</li> |
| 157 | + </ol> |
| 158 | + </section> |
| 159 | + |
| 160 | + <section aria-labelledby="real-analysis-section"> |
| 161 | + <h2 id="real-analysis-section">Real Analysis</h2> |
| 162 | + <ol class="topic-list"> |
| 163 | + <li><math><mi>σ</mi></math>-algebras of sets.</li> |
| 164 | + <li>Lebesgue measures and abstract measures, signed measures. Lebesgue-Stieltjes measures on the real line and their correspondence with increasing, right continuous functions.</li> |
| 165 | + <li>Measurable functions. Approximation by simple functions. Riemann and Lebesgue integrals.</li> |
| 166 | + <li>Monotone convergence and dominated convergence theorems, Fatou's lemma.</li> |
| 167 | + <li>Product spaces and product measure, Fubini-Tonelli theorems.</li> |
| 168 | + <li>Absolute continuity of measures, Radon-Nikodym theorem. Lebesgue-Radon-Nikodym decomposition.</li> |
| 169 | + <li>Hardy-Littlewood maximal function: the maximal inequality for the Hardy-Littlewood maximal function and the Vitali covering lemma. The Lebesgue differentiation theorem.</li> |
| 170 | + <li>Absolute continuity of functions, differentiation and the Fundamental Theorem of Calculus for absolutely continuous functions. The correspondence between absolutely continuous functions on <math><mi>ℝ</mi></math> and measures on <math><mi>ℝ</mi></math> which are absolutely continuous with respect to the Lebesgue measure. Bounded variation functions and their correspondence with complex measures on <math><mi>ℝ</mi></math>.</li> |
| 171 | + <li>Hölder's inequality, Jensen's inequality.</li> |
| 172 | + <li><math><msup><mi>L</mi><mi>p</mi></msup></math> spaces, completeness. Approximation of <math><msup><mi>L</mi><mi>p</mi></msup></math>-functions on <math><msup><mi>ℝ</mi><mi>d</mi></msup></math> by compactly supported continuous functions.</li> |
| 173 | + <li>Hilbert space, projection theorem, Riesz representation theorem, orthonormal sets, <math><msup><mi>L</mi><mn>2</mn></msup></math> spaces.</li> |
| 174 | + <li><math><msup><mi>L</mi><mi>p</mi></msup></math>-<math><msup><mi>L</mi><msup><mi>p</mi><mo>′</mo></msup></msup></math> duality when <math><mrow><mfrac><mn>1</mn><mi>p</mi></mfrac><mo>+</mo><mfrac><mn>1</mn><msup><mi>p</mi><mo>′</mo></msup></mfrac><mo>=</mo><mn>1</mn></mrow></math>.</li> |
| 175 | + <li>Elementary Fourier series, Riesz-Fischer and Parseval theorems. Riemann-Lebesgue theorem. Dirichlet kernel.</li> |
| 176 | + <li>Fourier transforms in <math><msup><mi>ℝ</mi><mi>d</mi></msup></math>, Plancherel and Parseval's theorems, Fourier transforms of derivatives and translations. Riemann-Lebesgue lemma. Hausdorff-Young's inequality.</li> |
| 177 | + <li>Convolutions: Fourier transforms of convolutions, approximations to the identity, approximation of functions on <math><msup><mi>ℝ</mi><mi>d</mi></msup></math> by compactly supported smooth functions. Young's inequality for convolution.</li> |
| 178 | + </ol> |
| 179 | + </section> |
| 180 | + |
| 181 | + <section class="references" aria-labelledby="references-section"> |
| 182 | + <h3 id="references-section">References</h3> |
| 183 | + <ul> |
| 184 | + <li>R. G. Bartle, <i>Elements of Real Analysis</i></li> |
| 185 | + <li>W. Rudin, <i>Principles of Mathematical Analysis</i></li> |
| 186 | + <li>H. L. Royden, <i>Real Analysis</i></li> |
| 187 | + <li>G. Folland, <i>Real Analysis</i></li> |
| 188 | + <li>S. Axler, <i>Measure, Integration & Real Analysis</i></li> |
| 189 | + <li>L. V. Ahlfors, <i>Complex Analysis</i></li> |
| 190 | + <li>J. B. Conway, <i>Functions of a Complex Variable</i></li> |
| 191 | + <li>T. Gamelin, <i>Complex Analysis</i></li> |
| 192 | + <li>D. Marshall, <i>Complex Analysis</i></li> |
| 193 | + <li>E. Stein and R. Shakarchi, <i>Complex Analysis</i></li> |
| 194 | + </ul> |
| 195 | + </section> |
| 196 | +</article> |
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