# Complex analysis exam 1

LaPlace Transform in Circuit Analysis Recipe for Laplace transform circuit analysis: 1. Redraw the circuit (nothing about the Laplace transform changes the types of elements or their interconnections). 2. Any voltages or currents with values given are Laplace-transformed using the functional and operational tables. 3.

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PRACTICE PROBLEMS FOR COMPLEX ANALYSIS 3 Problem 22: Let fbe a non-constant meromorphic function in C such that all poles of fare on the real line and are of the form nˇ, n2Z. Assume furthermore that for any x;y2R with y6= 0 it holds that jf(x+ iy)j ej yj(1 + 1 jyj) : Show that z= nˇis a pole of ffor any n2Z.

Final Exam 40% ; References: [A] Lars Ahlfors, Complex analysis. An introduction to the theory of analytic functions of one complex variable. Third edition. MR [S] Elias Stein and Rami Shakarchi, Complex Analysis. MR [N] Rolf Nevanlinna and Veikko Paatero, Introduction to complex analysis. Second Edition MR [C] Jonh Conway, Functions of one ...

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# Complex analysis exam 1

1.1 Complex Numbers and Their Properties 3 corresponding real and imaginary parts are equal.Since this simple concept is sometimes useful, we formalize the last statement in the next deﬁnition. Definition 1.2 Equality Complex numbers z 1 = a 1 + ib 1 and z 2 = a 2 + ib 2 are equal, z 1 = z 2,if a 1 = a 2 and b 1 = b 2.

1. Review of the properties of the argument of a complex number Before we begin, I shall review the properties of the argument of a non-zero complex number z, denoted by arg z (which is a multi-valued function), and the principal value of the argument, Arg z, which is single-valued and conventionally deﬁned such that: −π < Arg z ≤ π. (1) Analytic Functions of a Complex Variable 1 Deﬁnitions and Theorems 1.1 Deﬁnition 1 A function f(z) is said to be analytic in a region R of the complex plane if f(z) has a derivative at each point of R and if f(z) is single valued. 1.2 Deﬁnition 2 A function f(z) is said to be analytic at a point z if z is an interior point of some region ...
Example: Is A a subset of B, where A = {1, 3, 4} and B = {1, 4, 3, 2}? 1 is in A, and 1 is in B as well. So far so good. 3 is in A and 3 is also in B. 4 is in A, and 4 is in B. That's all the elements of A, and every single one is in B, so we're done. Yes, A is a subset of B. Note that 2 is in B, but 2 is not in A.

C. Complex Sentence D. Compound-Complex Sentence. 6. Bruno tends to get bored sitting on the beach, watching the waves, getting sand in his swimsuit, and reading detective novels for a week. A. Simple Sentence B. Compound Sentence C. Complex Sentence D. Compound-Complex Sentence. 7.

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