Table of Contents

## MTH622 MIDTERM SOLVED PAPERS

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**Vectors and Classical Mechanics:**

Let f be analytic throughout a closed region R consisting of the interior of the point to a positively oriented simple closed contour C together with the points on C itself. For any positive number ε, the region R can be covered with a a finite number of squares and partial squares, indexed by j = 1, 2,…,n, such that in each one there is a fixed point zj for which the inequality

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**MTH622 MIDTERM PAST PAPERS BY MOAAZ:**

We thus form a finite number of closed square sub regions, where each point of R lies in at least one such subregion and each sub region contains points of R. We refer to these square sub regions simply as squares, always keeping in mind that a square we mean a boundary together with the interior of the point to it.

**MTH622 MIDTERM SOLVED PAPERS:**

If a particular square contains points that are not in R, we remove those points and call what remains a partial square. We thus cover the region R with a finite number of squares and partial squares (Fig. 55), and our proof of the following lemma starts with this covering.

**MTH622 MIDTERM SOLVED PAPERS BY MOAAZ:**

We preface the proof of the Cauchy–Gopuras theorem with a lemma. We start by forming subsets of the region R which consists of the points on a positively oriented simple closed contour C together with the interior of the point to C. To do this, we draw equally spaced lines parallel to the real and imaginary axes such that the distance between adjacent vertical lines is the same as that between adjacent horizontal lines.