Complex analysis Definition and 784 Threads

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers. It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, thermodynamics, and particularly quantum mechanics. By extension, use of complex analysis also has applications in engineering fields such as nuclear, aerospace, mechanical and electrical engineering.As a differentiable function of a complex variable is equal to its Taylor series (that is, it is analytic), complex analysis is particularly concerned with analytic functions of a complex variable (that is, holomorphic functions).

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  1. arpon

    Complex Integration using residue theorem

    Homework Statement [/B] ##C_\rho## is a semicircle of radius ##\rho## in the upper-half plane. What is $$\lim_{\rho\rightarrow 0} \int_{C_{\rho}} \frac{e^{iaz}-e^{ibz}}{z^2} \,dz$$Homework Equations If ##C## is a closed loop and ##z_1, z_2 ... z_n## are the singular points inside ##C##...
  2. J

    I Can the Complex Integral Problem Be Solved Using Residue Theorem?

    I have this problem with a complex integral and I'm having a lot of difficulty solving it: Show that for R and U both greater than 2a, \exists C > 0, independent of R,U,k and a, such that $$\int_{L_{-R,U}\cup L_{R,U}} \lvert f(z)\rvert\,\lvert dz\rvert \leqslant \frac{C}{kR}.$$ Where a > 0, k...
  3. J

    I Problem with this estimation lemma example

    I have been trying to show that $$\lim_{U\rightarrow\infty}\int_C \frac{ze^{ikz}}{z^2+a^2}dz = 0 $$ Where $$R>2a$$ and $$k>0$$ And C is the curve, defined by $$C = {x+iU | -R\le x\le R}$$ I have tried by using the fact that $$|\int_C \frac{ze^{ikz}}{z^2+a^2}dz| \le\int_C...
  4. G

    A A problem about branch cut in contour integral

    Hello. I have a difficulty to understand the branch cut introduced to solve this integral. \int_{ - \infty }^\infty {dp\left[ {p{e^{ip\left| x \right|}}{e^{ - it\sqrt {{p^2} + {m^2}} }}} \right]} here p is a magnitude of the 3-dimensional momentum of a particle, x and t are space and time...
  5. dykuma

    Contour integral using residue theorem

    Homework Statement Find the solution of the following integral Homework Equations The Attempt at a Solution I applied the above relations getting that Then I was able to factor the function inside the integral getting that From here I should be able to get a solution by simply finding the...
  6. G

    I Domain of single-valued logarithm of complex number z

    Hello. Let's have any non-zero complex number z = reiθ (r > 0) and natural log ln applies to z. ln(z) = ln(r) + iθ. In fact, there is an infinite number of values of θ satistying z = reiθ such as θ = Θ + 2πn where n is any integer and Θ is the value of θ satisfying z = reiθ in a domain of -π <...
  7. Z

    I Proof check: S in C Compact implies S is closed and bounded

    I am using Lang's book on complex analysis, i am trying to reprove theorem 4.1 which is a simple theorem: Let Compact(S \in \mathbb{C}) \iff Closed(S) \land Bounded(S) I will show my attempt on one direction of the proof only, before even trying the other direction. Assume S is compact Idea...
  8. D

    Lower Bound on Weighted Sum of Auto Correlation

    Homework Statement Given ##v = {\left\{ {v}_{i} \right\}}_{i = 1}^{\infty}## and defining ## {v}_{n}^{\left( k \right)} = {v}_{n - k} ## (Shifting Operator). Prove that there exist ## \alpha > 0 ## such that $$ \sum_{k = - \infty}^{\infty} {2}^{- \left| k \right|} \left \langle {v}^{\left (...
  9. binbagsss

    Complex analysis f'/f , f meromorphic, Laurent series

    Homework Statement consider ##f## a meromorphic function with a finite pole at ##z=a## of order ##m##. Thus ##f(z)## has a laurent expansion: ##f(z)=\sum\limits_{n=-m}^{\infty} a_{n} (z-a)^{n} ## I want to show that ##f'(z)'/f(z)= \frac{m}{z-a} + holomorphic function ## And so where a...
  10. Alvis

    I Complex Analysis Harmonic functions

    Suppose u(x,y) and v(x,y) are harmonic on G and c is an element of R. Prove u(x,y) + cv(x,y) is also harmonic. I tried using the Laplace Equation of Uxx+Uyy=0 I have: du/dx=Ux d^2u/dx^2=Uxx du/dy=Uy d^2u/dy^2=Uyy dv/dx=cVx d^2v/dx^2=cVxx dv/dy=cVy d^2v/dy^2=cVyy I'm not really sure how to...
  11. F

    I Contour integration - reversing orientation

    I have been reading through "Complex Analysis for Mathematics & Engineering" by J. Matthews and R.Howell, and I'm a bit confused about the way in which they have parametrised the opposite orientation of a contour ##\mathcal{C}##. Using their notation, consider a contour ##\mathcal{C}## with...
  12. Pouyan

    Real part of a analytic function

    Homework Statement Show that xux + yuy is the real part of an analytic function if u(x,y) is. To which analytic function is the real part of u = Re (f(z))? Homework Equations What I know about analytic functions is Cauchy-Riemann condition (∂u/∂x) =(∂v/∂y) and (∂y/∂y)=-(∂v/∂x) I know...
  13. caffeinemachine

    MHB Stein-Shakarchi 'Complex Analysis' Chapter 3 Exercise 15 b.

    I am stuck on a problem given in Stein and Shakarchi's Complex Analysis. (Chapter 3, Exercise 15b) Use the maximum modulus principle or Cauchy inequalities to solve the following: Let $f$ be a bounded holomorphic function on the open unit disc, and suppose that $f$ converges uniformly to $0$ in...
  14. dumbdumNotSmart

    Complex Conjugate Inequality Proof

    Homework Statement $$ \left | \frac{z}{\left | z \right |} + \frac{w}{\left | w \right |} \right |\left ( \left | z \right | +\left | w \right | \right )\leq 2\left | z+w \right | $$ Where z and w are complex numbers not equal to zero. 2.$$\frac{z}{\left | z \right | ^{2}}=\frac{1}{\bar{z}}$$...
  15. Pouyan

    What could be causing delays in messages appearing on your screen?

    Consider the principal branch of the function f(z)= z7/3 Find f'(-i) and write it in the form a+bi My attemp is : I know zc = exp(c logz) and the derivative of that is : (c/z) * exp(c Logz) That is in this case (7/3)*(i) *exp((7/3)*Log-i) = f'(-i) I know that Log(-i) = Log(1) + i(-pi/2)= -i...
  16. Zeeree

    Mobius transformation for the first quadrant

    Homework Statement Find the images of the following region in the z-plane onto the w-plane under the linear fractional transformations The first quadrant ##x > 0, y > 0## where ##T(z) = \frac { z -i } { z + i }## Homework EquationsThe Attempt at a Solution [/B] So for this, I looked at the...
  17. Destroxia

    Complex Analysis - Exponential Form

    Homework Statement Write the given numbers in the polar form ##re^{i\theta}##. ## \frac {2i} {(3e^{4+i})} ## Homework Equations ## z = re^(i\theta) ## ## \theta = Arg(z) ## ## r = |z| = \sqrt { x^2 + y^2 } ## The Attempt at a Solution I'm not really sure how to go about the exponential...
  18. B

    Analysis Readability of Rudin's Real and Complex Analysis

    So I decide to self-study the real analysis (measure theory, Banach space, etc.). Surprisingly, I found that Rudin-RCA is quite readable; it is less terse than his PMA. Although the required text for my introductory analysis course was PMA, I mostly studied from Hairer/Wanner's Analysis by Its...
  19. Zeeree

    Finding an upper bound for a contour integral (Complex)

    C1 1. Homework Statement : Using the ML inequality, I have to find an upper bound for the contour integral of \int e^2z - z^2 \, dz where the contour C = C1 + C2. C1 is the circular arc from point A(sqrt(3)/2, 1/2) to B(1/2, sqrt(3)/2) and C2 is the line segment from the origin to B...
  20. B

    Analysis Supplements to Complex Analysis of Rudin-RCA?

    Dear Physics Forum friends, I will be doing a reading course in the complex analysis starting on this Fall Semester. The assigned book is Rudin's Real and Complex Analysis. From my understanding, Rudin treats complex analysis very elegantly, but very terse. I am curious if you could suggest...
  21. P

    How Is the Complex Integral Evaluated Using Contour Integration and Residues?

    Homework Statement calculate ## \int_{-\infty}^{\infty}{\frac{2}{1+x^2} e^{-ikx}dx}##, where k is any positive number Homework EquationsThe Attempt at a Solution So first consider the closed contour ##I= \int{\frac{2}{1+z^2} e^{-ikz}dx}## We can choose the contour to be along the real axis ##...
  22. K

    I Complex Analysis, holomorphic in circle.

    Hi, I have a question regarding corollary 2.3. in the uploaded image. it looks very trivial to me becauese Cauchy's theorem states "if f(z) is holomorphic, its closed loop integral will be always 0". Is this what the author is trying to say? what's the necesseity of the larger disk D' at here...
  23. K

    I Complex Analysis Radius of Convergence.

    Hello, I have two questions regarding the Radius of convergence. 1. What should we do at the interval (R-eps, R) 2. It used definition to prove radius of convergence, but I am not sure if it is necessary-sufficient condition of convergence. I get that this can be a sufficient condition but not...
  24. K

    I Complex Analysis holomorphic condition

    I understood the holomorphic condition this way. For a vector field F(x1, x2 . . ., xm) = <y1(x1, x2, x3 . . . , xm), y2(x1, x2, x3 . . . , xm), y3(x1, x2, x3 . . . , xm) . . . ,yn(x1, x2, x3 . . . , xm)> In a real analysis, its derivative is expressed as a Jacobian matrix because each...
  25. Jianphys17

    Which Complex Analysis Textbook Should I Use?

    Hi, i need advice, I'm studying complex analysis on which book would you recommend to do it on that of Ahlfors or T.W.Gamelin's book?
  26. P

    How Do You Evaluate sin(x)/x^4 Over the Unit Circle Using Complex Analysis?

    Homework Statement evaluate sinx/x^4 over the unit circle Homework Equations Cauchys Residue theorem ##sinz=1/(2i)(z+1/z)## The Attempt at a Solution So we have a branch point at z=0 but its of order 4 so I can't see any direct way of using Cauchys residue theorem. I've tried changing the...
  27. F

    Shading in Argand diagrams involving inequalities

    Homework Statement What is difference in shading between Argand diagrams containing inequalities with > and ≥ signs? Example Shade the appropriate region to satisfy the inequality |z|> 5 |z|≥ 5 The Attempt at a Solution I am aware of the fact that both will have circle centered at origin...
  28. Destroxia

    Why are complex numbers represented on a plane?

    So I know that a complex number can be represented by ##z=x+iy##, where ## z = x + iy \in \mathbb{C}##. Would it be okay to then state that ## z = x + iy \in \mathbb{C} := (x,y) \in \mathbb{R}^2 ##? If we can just look at complex numbers as coordinates in ##\mathbb{R}^2## what is the point of...
  29. hamad12a

    How to find the third root of z^3=1?

    Homework Statement in a given activity: solve for z in C the equation: z^3=1 Homework Equations prove that the roots are 1, i, and i^2 The Attempt at a Solution using z^3-1=0 <=> Z^3-1^3 == a^3-b^3=(a-b)(a^2+2ab+b^2) it's clear the solution are 1 and i^2=-1 but i didn't find "i" as a solution...
  30. A

    B Complex Replacement: Justification?

    Hi, Is there a proof that complex replacement is a valid way to solve a differential equation? I'm lacking some intuition on the idea that under any algebraic manipulations the real and imaginary parts of an expression don't influence each other. For example, if I'm given: $$p(D) x = cos(t)$$...
  31. P

    Complex Analysis: Contour Integration Question

    Homework Statement State, with justification, if the Fundamental Theorem of Contour Integration can be applied to the following integrals. Evaluate both integrals. \begin{eqnarray*} (i) \hspace{0.2cm} \int_\gamma \frac{1}{z} dz \\ (ii) \hspace{0.2cm} \int_\gamma \overline{z} dz \\...
  32. M

    Analysis Looking for a Good Book on Complex Analysis?

    Hi friends, I am looking for a good book on complex analysis. My goal is to add some new techniques to my problem solving arsenal, learn some elegant non-intuitive proofs and finally build up to knowing enough about the Zeta function to understand it's relationship with prime numbers.
  33. M

    B Limit using the Sohotski-Plemenj formula

    Hi All, I am desperate to understand a calculation presented in a paper by Sethna, "Elastic theory has zero radius of convergence", freely available online $$ lim_{\epsilon \to +0}Z(-P+i\epsilon) = lim_{\epsilon \to +0} \int_{0}^{\infty} \mathrm{d}x \, \int_{0}^{\infty} \mathrm{d}y \exp \{...
  34. Elvis 123456789

    How to represent this complex number?

    Homework Statement Finding "polar" and "rectangular" representation of a complex number? Make a table with three columns. Each row will contain three representations of a complex number z: the “rectangular” expression z = a + bi (with a and b real); the “polar” expression |z|, Arg(z); and a...
  35. matt_crouch

    How do I solve complex contour integrals in complex analysis?

    I am trying to teach myself complex analysis . There seems to be multiple ways of achieving the same thing and I am unsure on which approach to take, I am also struggling to visualise the problem...Would someone show me step by step how to solve for example...
  36. L

    Finding conditions that assure that a holomorphic function i

    Let H be a real-valued function of two real variables with continuous first partial derivatives. If h(z)=h(x+i y)= u(x,y)+iv(x,y)is holomorphic in a region V, and H(u,v)=0. Find a condition on H under which one can conclude that h is a constant . Of, course H is the zero function can do the job...
  37. L

    Geometric Interpretation of complex numbesr

    z1,z2,z3 are distinct complex numbers, prove that they are the vertices of an equilateral triangle if and only if the following relation is satisfied: z1^2+z2^2+z3^2=z1.z2+z2.z3+z3.z1 so i shall show that |z1-z2|=|z1-z3|=|z2-z3|but i do not know how to start.
  38. I

    Order of zero (complex analysis)

    I'm confused about how this related to the roots of an equation. Here's the definition: We say that ##f## has a zero of order ##m## at ##z_0## if ##f^{(k)}(z_0)=0## for ##k=0,\dots , m-1##, but ##f^{(m)}(z_0)\ne 0## or equivalently that ##f(z) = \sum_{k=m}^\infty a_k(z-z_0)^k##, ##a_m \ne 0##...
  39. EinsteinKreuz

    What is the mapping between the hyperriemann sphere and the complex plane?

    As many of you know, using the stereographic projection one can construct a homeomorphism between the the complex plane ℂ1 and the unit sphere S2∈ℝ3. But the stereographic projection can be extended to the n-sphere/n-dimensional Euclidean space ∀n≥1. Now what I am talking about is the the...
  40. SrVishi

    Analysis What are some alternatives to Rudin for learning complex analysis?

    Hello, I was wondering how well is Rudin's Real and Complex Analysis for learning complex analysis, assuming that difficulty won't be an issue. Does it cover the standard material? Is it deep enough? Should I just read from elsewhere instead?
  41. C

    Schools Is Complex Analysis a must for grad school applications?

    Is taking complex analysis before graduate school apps a "make-or-break" deal if one is looking to apply for theory? I am currently deciding whether to take it junior spring or defer it to senior spring. As it has come up in my research, I have studied some of it, but I'm wondering if it must be...
  42. sinkersub

    Inverse Binomial Expansion within Laurent Series?

    Homework Statement Find the Laurent Series of f(z) = \frac{1}{z(z-2)^3} about the singularities z=0 and z=2 (separately). Verify z=0 is a pole of order 1, and z=2 is a pole of order 3. Find residue of f(z) at each pole. Homework Equations The solution starts by parentheses in the form (1 -...
  43. matt_crouch

    Whittaker's solution and separable variables

    So It is well known that the 2D solution to the Laplace equation can be obtained by changing to complex coordinates ##u=x+iy## and ##v=x-iy##. This can be extended to n dimensions as long as the complex coordinates chosen also solve the Laplace equation. For example in 3D...
  44. J

    Why does (-1)^n appear in the power series for 1/(1+(z-1))?

    Homework Statement So I'm checking my solutions to past question and there's one bit that throws me. 1/(1+(z-1)) = Σ(-1)n(z-1)n (for 0<|z-1|<1) I don't know where the (-1)n factor came from. Is it just something that always happens that I didn't know about / forgot about, or is there some...
  45. V

    What is the "Book proof" of Euler's formula?

    The eccentric mathematician Paul Erdos believed in a deity known as the SF (supreme fascist). He believed the SF teased him by hiding his glasses, hiding his Hungarian passport and keeping mathematical truths from him. He also believed that the SF has a book that consists of all the most...
  46. M

    Singularities of a complex function

    Homework Statement [/B] Find and classify all singularities for (e-z) / [(z3) ((z2) + 1)] Homework EquationsThe Attempt at a Solution [/B] This is my first attempt at these questions and have only been given very basic examples, but here's my best go: I see we have singularities at 0 and i...
  47. J

    Complex Analysis: Conformal Mappings

    I am looking for conformal transformations to map: 1. Disk of radius R to equilateral triangular region with side A. 2. Disk of radius R to rectangular region with length L and width W. 3. Disk of radius R to elliptic disk with semi-major axis a and semi-minor axis b. Thanks!
  48. T

    Deciding between group theory vs. complex analysis

    Hi, I have one spot remaining to take a pure math course, and I'm trying to decide between complex analysis and group theory. Although I've touched some of the basic of dealing with complex numbers in my physics/DE courses, they haven't gone in much depth into them beyond applications. On the...
  49. thegirl

    Method used to find harmonic functions in complex analysis

    Hi, I was just wondering how would you go about finding a harmonic function in complex analysis when given certain conditions such as I am z > 0 and is 1 when x > 0 and 0 when x < 0. Do you draw a diagram? Do you solve the laplace equation? How would you go about doing this? What if there...
  50. J

    How can I find the first few terms of the Laurent series for

    Homework Statement For each of the following functions find the first few terms of each of the Laurent series about the origin, that is, one series for each annular ring between singular points. Find the residue of each function at the origin. The function is... 1/(z*(z-1)(z-2)^2) Homework...
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