## Berkeley problems in mathematicsIn 1977, the Mathematics Department at the University of California, Berkeley, instituted a written examination as one of the first major requirements toward the Ph.D. degree in mathematics. Its purpose was to determine whether first-year students in the Ph.D. program had successfully mastered basic mathematics in order to continue in the program with the likelihood of success. Since its inception, the exam has become a major hurdle to overcome in the pursuit of the degree. |

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### Contents

Real Analysis | 3 |

Multivariable Calculus | 29 |

Differential Equations | 39 |

Copyright | |

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abelian group analytic function assume automorphism basis bounded calculation shows characteristic polynomial coefficients commutative compact complex numbers constant continuous function continuous real valued contradiction converges uniformly cosets cyclic deﬁned denote diagonal diagonalizable differential equation dimension eigenvalues eigenvector element equal Evaluate exists f is continuous ﬁeld F ﬁnd ﬁnite ﬁnite-dimensional ﬁrst ﬁxed follows function f given group of order half-plane Hence homomorphism identity implies inequality inﬁnite integral interval invertible irreducible isomorphic Lemma Let f Let G linear transformation linearly maximum minimal polynomial multiplication nontrivial nonzero normal subgroup orthogonal permutations polynomial of degree positive deﬁnite positive integer prime Problem Prove that f radius of convergence real matrix real numbers real valued function Residue Theorem ring satisﬁes sequence Show that f Solution subgroup of G subset subspace tends to inﬁnity unique unit circle unit disc vector space zero