Complex analysis Definition and 756 Threads

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COPYRIGHT strictly reserved to Dr. T.E. Venkata Balaji (IIT Madras) and NPTEL, Govt of India. Duplication PROHIBITED.Lectures: http://www.nptel.ac.in/courses/111106094/Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111106094

COPYRIGHT strictly reserved to Dr. T.E. Venkata Balaji (IIT Madras) and NPTEL, Govt of India. Duplication PROHIBITED.Lectures: http://www.nptel.ac.in/courses/111106094/Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111106094

COPYRIGHT strictly reserved to Dr. T.E. Venkata Balaji (IIT Madras) and NPTEL, Govt of India. Duplication PROHIBITED.Lectures: http://www.nptel.ac.in/courses/111106094/Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111106094

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COPYRIGHT strictly reserved to Dr. T.E. Venkata Balaji (IIT Madras) and NPTEL, Govt of India. Duplication PROHIBITED.Lectures: http://www.nptel.ac.in/courses/111106094/Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111106094

COPYRIGHT strictly reserved to Dr. T.E. Venkata Balaji (IIT Madras) and NPTEL, Govt of India. Duplication PROHIBITED.Lectures: http://www.nptel.ac.in/courses/111106094/Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111106094

COPYRIGHT strictly reserved to Dr. T.E. Venkata Balaji (IIT Madras) and NPTEL, Govt of India. Duplication PROHIBITED.Lectures: http://www.nptel.ac.in/courses/111106094/Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111106094

COPYRIGHT strictly reserved to Dr. T.E. Venkata Balaji (IIT Madras) and NPTEL, Govt of India. Duplication PROHIBITED.Lectures: http://www.nptel.ac.in/courses/111106094/Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111106094

COPYRIGHT strictly reserved to Dr. T.E. Venkata Balaji (IIT Madras) and NPTEL, Govt of India. Duplication PROHIBITED.Lectures: http://www.nptel.ac.in/courses/111106094/Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111106094

COPYRIGHT strictly reserved to Dr. T.E. Venkata Balaji (IIT Madras) and NPTEL, Govt of India. Duplication PROHIBITED.Lectures: http://www.nptel.ac.in/courses/111106094/Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111106094

COPYRIGHT strictly reserved to Dr. T.E. Venkata Balaji (IIT Madras) and NPTEL, Govt of India. Duplication PROHIBITED.Lectures: http://www.nptel.ac.in/courses/111106094/Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111106094

COPYRIGHT strictly reserved to Dr. T.E. Venkata Balaji (IIT Madras) and NPTEL, Govt of India. Duplication PROHIBITED.Lectures: http://www.nptel.ac.in/courses/111106094/Syllabus: http://www.nptel.ac.in/syllabus/syllabus.php?subjectId=111106094

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If a Laplace transform has a region of convergence starting at Re(s)=0, does the Laplace transform evaluated at the imaginary axis exist? I.e. say that the Laplace transform of 1 is 1/s. Does this Laplace transform exist at say s=i?

Hey everyone 1. Homework Statement I want to compute the Taylor expansion (the first four terms) of $$f(x) =x/sin(ax)$$ around $$x_0 = 0$$. I am working in the space of complex numbers here. Homework Equations function: $$f(x) = \frac{x}{\sin (ax)}$$ Taylor expansion: $$ f(x) = \sum...

Homework Statement Identify the set of points satisfying ##1<\vert 2z-6\vert <2## such that ##z\in\Bbb{C}##. My pre-caculus is very rusty, so I am not sure if I am doing this correctly. Homework Equations ##x^2 +y^2= r^2## ##\forall z,z'\in\Bbb{C}, \vert zz'\vert =\vert z\vert\vert z'\vert##...

The problem I am trying to calculate the integral $$ \int_{\gamma} \frac{z}{z^2+4} \ dz $$ Where ## \gamma ## is the line segment from ## z=2+2i ## to ## z=-2-2i ##. The attempt I would like to solve this problem using substitution and a primitive function to ## \frac{1}{u} ##. I am not...

The problem I would like to solve: $$ \bar{z} = z^n $$ where ##n## is a positive integer. The attempt ## z = r e^{i \theta} \\ \\ \overline{ r e^{i \theta} } = r^n e^{i \theta n} \\ r e^{-i \theta} = r^n e^{i \theta n} ## ## r = r^n \Leftrightarrow true \ \ if \ \ n=1 \ \ or \ \ r=1## ##...

Suppose f,g:ℂ→ℂ are analytic with singularities at z=0. I was wondering whether f(z)^2 or f(z)g(z) will have a singularity at z=0? For each, can you give me a proof or a counterexample?

Hello everyone! I'm having a bit of a problem with comprehension of the Cauchy integration formula. I might be missing some key know-how, so I'm asking for any sort of help and/or guideline on how to tackle similar problems. I thank anyone willing to take a look at my post! Homework Statement...

Homework Statement Find Laurent series of $$z^2sin(\frac{1}{1-z})$$ at $$0<\lvert z-1 \rvert<\infty$$ Homework Equations sine series expansion. The Attempt at a Solution At first, it seems pretty elementary since you can set w=\frac{1}{z-1} and expand at infinity in z, which is 0 in w...

Homework Statement Use Cardano's formula to find a real root for ##3x^3-45x^2+243x-525=0##. [Edited to correct mistake] Homework Equations $$x = u - \frac{b}{3a}$$ Depressed cubic: $$u^3=3pu+2q$$ Cardano's formula: $$u=\sqrt[3]{q+\sqrt{q^2-p^3}}+\sqrt[3]{q-\sqrt{q^2-p^3}}$$ The Attempt at a...

Hello, I would like your help understanding how to map a region of the space \mathbb{C}^2 spanned by two complex conjugate variables to the real plane \mathbb{R}^2 . Specifically, let us think that we have two complex conugate variables z and \bar{ z} and we define a triangle in the...

I am reading Bruce P. Palka's book: An Introduction to Complex Function Theory ... I am focused on Chapter 4: Complex Integration, Section 1.2 Smooth and Piecewise Smooth Paths ... I need help with some aspects of Example 1.3, Section 1.2, Chapter 4 ... Example 1.3, Section 1.2, Chapter 4...

Homework Statement What is the argument of -4-3i, and -4+3i? Homework Equations tantheta=opposite/adjacent side The principle of argument is that the argument lies between -pi and pi (not including -pi). The Attempt at a Solution arg(-4-3i) = -pi + arctan(3/4) arg(-4+3i) = pi - arctan(3/4)...

I saw the sentence " So the contour integral of an analytic function f(z) around a tiny square of size e is zero to order e^2. ". I want to know what " be zero to order " means exactly.

Does a circular function with complex variable represent a three-dimensional graph? For example cosiz

I´m having a problem with the value of the expression ##F(it)-F(-it)##, found on the Abel-Plana formula, where $$F(z)=\sqrt{z^2 + A^2}$$, with ##A## being a positive real number (F(z) is analytic in the right half-plane). Well, I know the result is ##F(it)-F(-it)=2i\sqrt{t^2 -A^2}##, for...

What book do MHB members regard as the best for a rigorous but clear and (moderately) easily understood introduction to complex analysis? (Note - would be good if the book had hints to solutions of exercise.) Peter

https://arxiv.org/pdf/1705.07188.pdf Equation 5 in this paper states that $$\frac{\partial F}{\partial p_i} = 2Re\left\lbrace\frac{\partial F}{\partial x}\frac{\partial x}{\partial p_i}\right\rbrace$$ Here, p_i stands for the i'th element of a vector of 'design parameters' \mathbf{p}. These...

I'm looking for a good Complex book, but the options seem slim. I was thinking about Rudin's Real and Complex. My only reservation is that it is not structured like any other book I've seen. I've had advanced analysis and measure and integration theory, so rigour is not a concern. I saw Alfohr's...

Hello! I have this Proposition: "A harmonic function is infinitely differentiable". The book gives a proof that uses this theorem: "Suppose u is harmonic on a simply-connected region G. Then there exists a harmonic function v in G such that ##f = u + iv## is holomorphic in G. ". In the proof...

Homework Statement Suppose f is entire and there exist constants a and b such that ##|f(z)| \le a|z|+b## for all ##z \in C##. Prove that f is a polynomial of degree at most 1. Homework EquationsThe Attempt at a Solution We have that for any ##z \neq 0##, ##\frac{|f(z)|}{a|z|} \le b##. So if we...

Homework Statement $$\int_{-\infty}^\infty \space \frac{cos(2x)}{x-3i}dx$$ Homework EquationsThe Attempt at a Solution $$\int_{-R}^R \space \frac{e^{2ix}}{x-3i}dx + \int_{C_R} \space \frac{e^{2iz}}{z-3i}dz = 2\pi i\sum\space res \space f(z)$$ Then using Jordan's Lemma, as ##R\to\infty## the...

Homework Statement Compute the Integral: ##\int_{-\infty}^\infty \space \frac{e^{-2ix}}{x^2+4}dx## Homework Equations ##\int_C \space f(z) = 2\pi i \sum \space res \space f(z)## The Attempt at a Solution At first I tried doing this using a bounded integral but couldn't seem to get the right...

Homework Statement Question asks to show that if f is an entire function and bounded then it is polynomial of degree m or less. Homework Equations The Attempt at a Solution I tried plugging in the power series for f(z) and tried/know it is related to Liouville's Theorem somehow but I am...

<Moderator's note: moved from a technical forum, so homework template missing> Hi. I have solved the others but I am really struggling on 22c. I need it to converge for |z|>2. This is the part I am really struggling with. I am trying to get both fractions into a geometric series with...

Homework Statement Our textbook, Fundamentals of Complex Analysis, (...) by Saff Snider says on page 135 that by choosing some suitable branch for the square root and the logarithm then one can show that any such branch satisfies the equation below. The homework/task is to find all such branch...