finalexamproblems-1-1

finalexamproblems-1-1 - Math 322 Spring 2008 Review...

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Math 322. Spring 2008 Review Problems For The Final Exam Topics for midterm I & II Topic 1: Complex Numbers SpeciFc topics: Polar form of complex number; Operations of complex num- bers; Roots of complex number; Continuity, Differentiability and analyticity of complex functions; Cauchy-Riemann equations; harmonic function and har- monic conjugate; Exponential, trigonometric, hyperbolic and logarithmic func- tions, general power. Problem 1.1 : Let z 1 =2 - 2 i , z 2 = 2 + 3 i , fnd (a.) z 1 + z 2 ¯ z 2 2 (b.) Im ([(1 - i ) 8 z 2 1 ]) (c.) | z 1 - z 2 z 2 | (d.) Re (( z 1 + 1) z 2 ) Problem 1.2 : Find all the solutions ±or z 3 =1 . Problem 1.3 : Find out whether the ±ollowing ±unction is continuous at z =0 . f ( z )= ± Im ( z ) 1 -| z | ,z ± ; 0 . Problem 1.4 : Use Cauchy-Riemann equations to check whether the ±ollowing ±unc- tion is analytic. f ( z )=2 z - z Problem 1.5 : Veri±y that u ( x, y x 3 - 3 xy 2 - 2 x is harmonic in the whole com- plex plane, fnd a harmonic conjugate ±unction v ( x, y ) u ( x, y ) , and fnd f ( z u ( x, y )+ iv ( x, y ) . Problem 1.6 : Compute sin(2 + 3 i ) , e - 3+2 i , cosh(3 + nπi ) , Ln ( - 3+ i ) . Write your answer in the ±orm a + bi .
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Topic 2: Linear Algebra Specifc topics: Matrix Operations; Linear Independence oF vectors; Linear system oF equations; Rank, row space, column space, basis; Determinant, In- verse; Eigenvalues and eigenvectors. Problem 2.1 : Let A= - 12 01 50 , B= ± 2 1 3 5 - ² Calculate the following products or sums or give reasons why they are not deFned.
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finalexamproblems-1-1 - Math 322 Spring 2008 Review...

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