By Ravi P. Agarwal, Kanishka Perera, Sandra Pinelas

ISBN-10: 1461401941

ISBN-13: 9781461401940

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

Key positive aspects of this textbook:

-Effectively organizes the topic into simply conceivable sections within the type of 50 class-tested lectures

- makes use of certain examples to force the presentation

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

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

-Provides a concise heritage of advanced numbers

An advent to advanced research might be worthy to scholars in arithmetic, engineering and different technologies. necessities comprise a direction in calculus.

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Extra info for An Introduction to Complex Analysis

Example text

Thus, if and only if Arg z1 -Arg z2 = 2kπ, k ∈ Z. But for −π < Arg z1 , Arg z2 ≤ π, the only possibility is Arg z1 = Arg z2 . Conversely, if Arg z1 = Arg z2 , then Re (z1 z 2 ) = r1 r2 = |z1 ||z2 |. Now, |z1 + z2 | = |z1 | + |z2 | ⇐⇒ z1 z 1 + z2 z 2 + z1 z 2 + z2 z 1 = |z1 |2 + |z2 |2 + 2|z1 |z2 | ⇐⇒ z1 z 2 + z2 z 1 = 2|z1 ||z2 | ⇐⇒ Re(z1 z 2 + z2 z 1 ) = Re(z1 z 2 ) + Re(z2 z 1 ) = 2|z1 ||z2 | ⇐⇒ Re(z1 z 2 ) = |z1 ||z2 | and Re(z 1 z2 ) = |z1 ||z2 | ⇐⇒ Arg (z1 ) = Arg √ (z2 ). 15. If a is a positive real number, then a denotes the positive square root is the meaning of √ of a.

Hence, deduce that the function w = f (z) = ze−|z| is not analytic. 12. Let w = f (z) be analytic in a neighborhood of z0 , and w0 = f (z0 ), f (z0 ) = 0. Show that f defines a one-to-one mapping of a neighborhood of z0 onto a neighborhood of w0 . 13. Use L’Hˆopital’s Rule to find the following limits: z7 + i z 4 − 81 , (b). lim , z→i z 14 + 1 z→3i z 2 + 9 (a). lim (c). lim √ 1+i z 6 − 64 . 14. , independent of x and y), then f is a constant. 15. Show that if h : IR2 → IR and f = 2h3 + ih is an entire function, then h is a constant.

C, where z0 = z1 and c = 1, z − z1 (g). 0 < Im z < 2π, Re z (h). > 1, Im z < 3, |z − 1| (i). |z − z1 | + |z − z2 | = 2a, (j). azz + kz + kz + d = 0, k ∈ C, a, d ∈ IR, and |k|2 > ad. 3. Let α, β ∈ C. Prove that |α + β|2 + |α − β|2 = 2(|α|2 + |β|2 ), and deduce that |α + α2 − β 2 | + |α − α2 − β 2 | = |α + β| + |α − β|. 4. Use the properties of conjugates to show that (a). (z)4 = (z 4 ), (b). z1 z2 z3 = z1 . 5. If |z| = 1, then show that az + b bz + a = 1 16 Lecture 3 for all complex numbers a and b.

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