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Complex Analysis Lecture Notes

1 Complex analysis is in my opinion one of the most beautiful areas of mathemat-ics It has one of the highest ratios of theorems to de nitions (i e a very low \\entropy\") and lots of applications to things that seem unrelated to complex numbers for example: Solving cubic equations that have only real roots (historically this was the

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Complex Analysis Lecture Notes

These notes are about complex analysis the area of mathematics that studies analytic functions of a complex variable and their properties While this may sound a bit specialized there are (at least) two excellent reasons why all mathematicians should learn about complex analysis First it is in my humble opinion one of the most beautiful areas

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COMPLEX ANALYSIS: LECTURE NOTES

COMPLEX ANALYSIS: LECTURE NOTES COMPLEX ANALYSIS: LECTURE NOTES DMITRI ZAITSEV Contents 1 The origin of complex numbers 4 1 1 Solving quadratic equation 4 1 2 Cubic equation and Cardano’s formula 4 1 3 Example of using Cardano’s formula 5 2 Algebraic operations for complex numbers 5 2 1 Addition and multiplication 5 2 2

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Lectures on complex analysis

In this ﬁrst chapter I will give you a taste of complex analysis and recall some basic facts about the complex numbers We deﬁne holomorphic functions the subject of this course These functions turn out to be much more well-behaved than the functions you have encountered in real analysis We will mention the most striking such properties

What is complex analysis?
These notes are about complex analysis, the area of mathematics that studies analytic functions of a complex variable and their properties. While this may sound a bit specialized, there are (at least) two excellent reasons why all mathematicians should learn about complex analysis.
What is the most important integral in complex analysis?
Let us next consider the most important integral in complex analysis. It concerns the holomorphic function = 2πi. This doesn’t contradict the miraculous property discussed in the first lecture, which says that integrals along homotopic paths are equal: the circle can’t be contracted within the domain Ω, which has a hole at the origin.
Which textbook is used in the exposition of complex analysis?
With some exceptions, the exposition follows the textbook Complex Analysis by E. M. Stein and R. Shakarchi (Princeton University Press, 2003). Acknowledgements. I am grateful to Christopher Alexander, Jennifer Brown, Brynn Caddel, Keith Conrad, Bo Long, Anthony Nguyen, Jianping Pan, and Brad Velasquez for comments that helped me improved the notes.

1 Introduction: why study complex analysis?

These notes are about complex analysis, the area of mathematics that studies analytic functions of a complex variable and their properties. While this may sound a bit specialized, there are (at least) two excellent reasons why all mathematicians should learn about complex analysis. First, it is, in my humble opinion, one of the most beautiful areas

3 Analyticity, conformality and the Cauchy-Riemann equa-tions

In this section we begin to build the theory by laying the most basic cornerstone of the theory, the de nition of analyticity, along with some of the useful ways to think about this fundamental concept. math.ucdavis.edu

4 Power series

Until now we have not discussed any speci c examples of functions of a complex variable. Of course, there are the standard functions that you probably encountered already in your undergraduate studies: polynomials, rational functions, ez, the trigonometric functions, etc. But aside from these examples, it would be useful to have a general way to co

6 Cauchy's theorem

One of the central results in complex analysis is Cauchy's theorem. Theorem 6 (Cauchy's theorem.). If f is holomorphic on a simply-connected region curve in we have I f(z) dz = 0: , then for any closed The challenges facing us are: rst, to prove Cauchy's theorem for curves and regions that are relatively simple (where we do not have to deal with su

w z w z

Integrating each term separately, we have for the rst term math.ucdavis.edu

; ). This is called the principal branch of the logarithm basically

a kind of standard version of the log function that complex analysts have agreed to use whenever this is convenient (or not too inconvenient). However, sometimes we may want to consider the logarithm function on more strange or complicated regions. When can this be made to work? The answer is: precisely when is simply-connected. math.ucdavis.edu

14 The Euler gamma function

The Euler gamma function (often referred to simply as the gamma function) is one of the most important special functions in mathematics. It has applications to many areas, such as combinatorics, number theory, di erential equations, probability, and more, and is probably the most ubiquitous transcendental function after the \\elementary" transcenden

15.1 De nition and basic properties

The Riemann zeta function (often referred to simply as the zeta function when there is no risk of confusion), like the gamma function is considered one of the most important special functions in \\higher" mathematics. However, the Riemann zeta function is a lot more mysterious than the gamma function, and remains the subject of many famous open prob

15.2 A theorem on the zeros of the Riemann zeta function

Next, we prove a nontrivial and very important fact about the zeta function that will play a critical role in our proof of the prime number theorem. math.ucdavis.edu

(x) x.

Proof. Note the inequality (x) = X log p p x log x log p math.ucdavis.edu

(x) x= log x.

Now assume that (x) x logx, and apply the inequalities we derived above in the opposite direction from before. That is, we have math.ucdavis.edu

17 Introduction to asymptotic analysis

In this section we'll learn how to use complex analysis to prove asymptotic formulas such as n p(n) math.ucdavis.edu

0 = e xxn x 1 = e xn + nxn = e xxn 1( x + n); dx

i. when x = n. Plugging this value into the inequality gives the bound n (n=e)n (n 1): 3As a general rule of problem-solving, it's often helpful to start attacking a problem by thinking about really easy things you can say about it before moving on to advanced techniques. It's a great way to develop your intuition, and sometimes you discover that

17.3 A conceptual explanation

In both the examples of Stirling's formula and the central binomial coe cient we analyzed above, we made what looked like ad hoc choices regarding how to \\massage" the integrals, what value r to use for the radius of the contour of integration, what change of variables to make in the integral, etc. Now let us think more conceptually and see if we c

Problems

Below is a list of basic formulas in complex analysis (think of them as \\formulas you need to know like the back of your hand"). Review each of them, making sure that you understand what it says and why it is truethat is, if it is a theorem, prove it, or if it is a de nition, make sure you understand that that is the case. In the formulas below, a

k. w z = w z

l. jwzj = jwj jzj m. jwj j zj jw + zj jwj + jzj n. ex+iy = ex(cos(y) + i sin(y)) math.ucdavis.edu

15. Cauchy's theorem and irrotational vector

elds. Recall from vector calculus that a planar vector eld F = (P; Q) de ned on some region C = R2 is called conservative if it is of the form @g F = rg = @x; @g (the gradient of g) for some scalar function g : @y theorem of calculus for line integrals, for such a vector eld we have R. By the fundamental I math.ucdavis.edu

(s) = ( s)

of the gamma function, also considered as a somewhat important special function in its own right. (a) Show that (s) has the convergent series expansions (s) = math.ucdavis.edu

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