What is Complex variables: Definition and 120 Discussions

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. F

    Complex variables questions, limits and bounds

    If I have a function f(z) such that limit z --> infinity is finite what do I know about that function? Is it "bounded" or am I still confused about what "bounded" means?
  2. Simfish

    Integral of 1/x in complex variables

    Does \int 1/x dx = ln|x| + c in complex variable theory? Or can we relax the absolute value restraint of ln|x|? (as in, can it be ln(x) + c?)
  3. M

    Proving well defined (complex variables)

    Homework Statement Let D be a connected domain in R^2 and let u(x,y) be a continuous vector field defined on D. Suppose u has zero circulation and zero flux for any simple closed contour on D. u(x,y) = (u_1(x,y),u_2(x,y)) \Gamma = \int_{c}(u\circ\gamma)tds = 0F=\int_{c}(u\circ\gamma)nds =...
  4. M

    Proving well defined (complex variables)

    Homework Statement Let D be a connected domain in [tex]R^2[\tex] and let u(x,y) be a continuous vector field defined on D, [tex]u(x,y) = (u_1(x,y),u_2(x,y))[\tex][tex]\Gamma = \int_{c}(u\circ\gamma)tds[\tex] [tex]F=\int_{c}(u\circ\gamma)nds[\tex] Homework Equations The Attempt at a Solution...
  5. F

    I think I just flunked the in-class portion of my complex variables

    I think I just flunked out on the in-class portion of my complex variables midterm. I have a take home exam to work on over the weekend, and it's not going to be easy. I studied for days for this midterm, the prof said it would be on "proofs" so I learned every proof in the book. But, I wish I'd...
  6. F

    Complex variables: ML inequality

    I'm studying the proof of the ML inequlity from complex analysis. I don't know what they did in one step of the proof and I was wondering if anyone can explain the step to me. First of all the theorem says: Let f(z) be continuous on a contour C. Then |\int_{c}f(z)dz| \leq ML Where L...
  7. D

    Constructing a Bounded Non-Convergent Sequence in Complex Variables

    I can't think of how to title the problem I'm having, but this is what the course is called. Complex being imaginary numbers, ie z = a + ic where i is the sqrt of -1. So here is the question that I have no idea where to start with: Construct a sequence {zn} which is bounded and for...
  8. B

    Why Does the Complex Integral Not Represent Area Like in Real Calculus?

    Hi. I am studying on complex variables on my own using Brown and Churchill, 6th edition. so far i am doing pretty good, but i have run into trouble doing a few contour integrals. 1. First off, the authors state that unlike in calculus, where the integral can mean area under the curve, in...
  9. L

    How to derive the formula for tan(-1)(z) ~ Complex variables

    I have to prove that tan^(-1)(z) = i/2 * log[(i+z)/(i-z)] Homework Equations sin^(-1)(z) = -i * log[iz + (1-z^2)^(1/2)] cos^(-1)(z) = -i * log[z + i(1-z^2)^(1/2)] The Attempt at a Solution I've tried a number of attemps. I first tried the most obvious and I did sin^(-1)(z)...
  10. S

    Complex Variables Derivation of arcsine derivative

    Greetings, I'm working (playing) on a problem involving approximating the arcsin() function. I've attmpted to verify the known derivative of the arcsin function (d(arcsin))/dx = 1/sqrt(1-z^2) I know I have a mistake in my derivation. I've attached an electronic copy of my work...
  11. S

    How to solve two equations involving complex variables?

    how to solve two equations involving complex variables?? Hi guys! well! i want to know how can u solve the following two equations simultaenously in order to find out Ix and Iy: (3+j4)Ix - j4Iy=10-------------(1) (2-j4)Ix +j2Iy=0---------------(2) please tell me the best methods to...
  12. A

    What is a good intro to complex variables book for an undergrad?

    I'm wondering if anyone can recommend a good intro to complex variables book for an undergrad (calc1-3,diffeq). Preferably something that can be found cheap on amazon (Dover?)
  13. I

    (Complex Variables) Differentiability of Arg z

    I am proving that the function f(z) = Arg z is nowhere differentiable by using the definiton of a derivative. I let z = x + yi. Then, if the limit exists, we have f'(z) = lim (/\z -> 0) ( f(z + /\z) - f(z) ) / /\z. (Note that /\ is the triangle symbol) Also, let /\z = p + iq, where p and...
  14. N

    Which is more important: Partial DiffE or Complex Variables

    I am majoring in EE and have the option of taking either of these classes. I have to take Calc 1-3, Ordinary DiffE, and Linear Algebra. So after these classes is PDE or Complex Variables more useful to electrical engineers?
  15. A

    Number Theory or Complex Variables?

    Hey all, great site and I look forward to contributing more. For now, a question... For next semester I need to choose between Number Theory or Complex Variables. I am under the impression that complex variables will be the more useful class for my physics education, however I had some concerns...
  16. N

    Need some help with basic complex variables (no proofs)

    need some urgent help with basic complex variables (no proofs) Hi: can someone give me examples of the following? (no proofs needed) 1. a non-zero complex number z such that Arg(z^2) "not equal to" 2 Arg z 2. a region in C which is not a domain 3. a non-empty subset of C which has no...
  17. D

    Applications of Complex Variables

    I just signed up to take Applications of Complex Variables next term and wondering if anyone here has the scoop on it. Mainly, what are the main topics and any advice on the class or important parts I should really pay attention to.
  18. E

    Complex Variables and Integration

    How do I solve the following integral using complex variable techniques The integral from 0 to infinity of [x^m/(x^2 + 1)^2]; 1<m<3
  19. E

    Evaluation of Integrals using Complex Variables

    So how would one solve the following integral using complex variables?, Integral from 0 to pi of [dx/r +5cos(x)] where 0<r<5
  20. H

    Complex Variables: Questions on Singularities, Residues & Cauchy's P.V.

    I have several questions on complex variables, so I will just put them all in here. 1. What are the positions and natures of the singularities and the residues at the singularities of the following functions in the z-plane, excluding the point at infinity...
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