By Ravi P. Agarwal, Kanishka Perera, Sandra Pinelas

This textbook introduces the topic of complicated research to complicated undergraduate and graduate scholars in a transparent and concise manner.

Key gains of this textbook:

-Effectively organizes the topic into simply achievable sections within the kind of 50 class-tested lectures

- makes use of specific examples to force the presentation

-Includes a variety of workout units that inspire pursuing extensions of the fabric, every one with an “Answers or tricks” part

-covers an array of complicated subject matters which enable for flexibility in constructing the topic past the fundamentals

-Provides a concise historical past of complicated numbers

An creation to complicated research can be priceless to scholars in arithmetic, engineering and different technologies. must haves contain a direction in calculus.

**Read or Download An Introduction to Complex Analysis PDF**

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**Additional resources for An Introduction to Complex Analysis**

**Sample text**

If limz→z0 f (z) = A and limz→z0 g(z) = B, then limz→z0 (f (z) ± g(z)) = A ± B, (ii) limz→z0 f (z)g(z) = AB, and f (z) A (iii) limz→z0 = if B = 0. g(z) B (i) Complex Functions 31 For the composition of two functions f and g denoted and deﬁned as (f ◦ g)(z) = f (g(z)), we have the following result. 3. If limz→z0 g(z) = w0 and limw→w0 f (w) = A, then lim f (g(z)) = A = f z→z0 lim g(z) . z→z0 Now we shall deﬁne limits that involve ∞. For this, we note that z → ∞ means |z| → ∞, and similarly, f (z) → ∞ means |f (z)| → ∞.

8. Let f = u + iv, where u = x2 In spite of the two examples above, we have the following result. 5 (Suﬃcient Conditions for Diﬀerentiability). Let f (z) = u(x, y) + iv(x, y) be deﬁned in some open set S containing the point z0 . If the ﬁrst order partial derivatives of u and v exist in S, are continuous at z0 , and satisfy the Cauchy-Riemann equations at z0 , then f is diﬀerentiable at z0 . Moreover, f (z0 ) = = ∂u ∂v (x0 , y0 ) + i (x0 , y0 ) ∂x ∂x ∂v ∂u (x0 , y0 ) − i (x0 , y0 ). ∂y ∂y Analytic Functions I 41 Consequently, if the ﬁrst-order partial derivatives are continuous and satisfy the Cauchy-Riemann equations at all points of S, then f is analytic in S.

Suppose that f : U → C is continuous and U is compact. Consider a covering of f (U ) to be open sets V. The inverse images f −1 (V ) are open and form a covering of U. 4 we can select a ﬁnite subcovering such that U ⊂ f −1 (V1 ) ∪ · · · ∪ f −1 (Vn ). 4 implies that f (U ) is compact. (ii). Suppose that f : U → C is continuous and U is connected. If f (U ) = A ∪ B where A and B are open and disjoint, then U = f −1 (A) ∪ f −1 (B), which is a union of disjoint and open sets. Since U 36 Lecture 5 is connected, either f −1 (A) = ∅ or f −1 (B) = ∅, and hence either A = ∅ or B = ∅.