Complex variables Definition and 120 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. E

    Derivative of Complex Variables

    Hi, What is the following derivative: \frac{\partial}{\partial x}|b-ax|^2? Now I know that |b-ax|^2=(b-ax)(b^*-a^*x^*), so how to do the differentiation with respect to x^*? Thanks in advance PS.: All variables and constants are complex.
  2. N

    What are some good websites that'll help me understand Complex Variables?

    The lectures haven't been difficult, but the book is hard for me to read. I usually understand math by working problems, but this course doesn't seem to work like that. I just don't know how to complete the exercises in the book. The book is "An Introduction to Complex Analysis and Geometry"...
  3. S

    Complex variables- graphing an equation

    Homework Statement Suppose that c is a member of the Real numbers, and p is a member of the Complex numbers with p not equal to 0, are given numbers. (a) Show that pz + conjugate(pz) + c = 0 is the equation of a straight line in the plane. Provide a carefully-drawn plot that...
  4. K

    Courses ADVICE on complex variables course needed:

    i'm sorry if this is the wrong area to be posting such a question, but I am registered for a complex analysis course starting in the fall and I was wondering what i should review before starting this course? I've had math coursework up through calculus, differential equations, linear algebra...
  5. G

    Real integral using complex variables

    Homework Statement \int_{0}^{\infty }\frac{dx}{1+x^{5}} The attempt at a solution This is for my complex variables class, so I have been trying to compute it using residues. I noticed that if we extend it to the complex plane and integrate over the edges of a half disk or a quarter...
  6. E

    Mapping and Rotation of Complex Curves

    Homework Statement Find the angle through which a curve drawn from the point z0 is rotated under the mapping w=f(z), and find the corresponding scale factor of the transformation. z0 = -1, w=z^2 and z0 = -1 + i, w = 1/z Homework Equations I honestly don't know how to begin...
  7. M

    Definition of a limit with complex variables

    Homework Statement Use the definition of a limit to show that lim (z^{2} +c) = z_{0}^{2} +c as {z->z_{0}}Homework Equations Definition of a limit: |f(z)-L|< epsilon if 0<|z-z0|< delta The Attempt at a Solution |(z^{2}+c )-(z_{0}^{2}+c)| = | z^{2}-z_{0}^{2}|= |(z-z_{0})(z+z_{0}) | <...
  8. P

    Complex Variables Homework Solutions

    Homework Statement http://www.shotpix.com/images/83975343306312755574.jpg Homework Equations The Attempt at a Solution For queation 1, I've found w = u + jv = sinxcoshy + jcosxsinhy u= sinxcoshy v=cosxsinhy Then I put the 4 boundary line into sin(z), I get y=0 ==> w=sinx...
  9. J

    What Is a Complex Polynomial Differentiable Only on the Unit Circle?

    Find a (complex) polynomial function f of x and y that is differentiable at the origin, with df/dz = 1 at the point z=0, and differentiable at all points on the unit circle x^2 + y^2=1, but is not differentiable at any other point in the complex plane. (Bruce Palka, Page 101) I think we use...
  10. J

    How Do You Convert sin^2(z) into x+iy Form?

    Homework Statement How do I change sin^2(z) to x+iy form? (z=x+iy) I have to put this x and y to arctan(y/x) Homework Equations The Attempt at a Solution I tried to use sin^2(z) = 1/2 -1/2(cos(2z)) or sin(z) = ((e^(iz) - e^(-iz))/2i)^2 but both ways I cannot take out i. Or isn't the...
  11. M

    Solving PDEs involving complex variables

    Hello all, I was wondering if you could share your thoughts regarding how one should go about solving PDEs in which all or some of the variables are complex. To solve ODEs involving real variables, my favorite method is to take the equations to Laplace domain, then solving the resulting set of...
  12. J

    Complex Variables: Proving 1+2w+3w^2+...+nw^(n-1) = n/(w-1)

    Homework Statement Let w = e^((2pi*i)/n). Show that 1+2w+3w^2+...+nw^(n-1) = n/(w-1) Homework Equations 1+x+x^2+x^3+...+x^m = (1-x^(m+1))/(1-x) --> A 1+x+x^2+x^3+... = 1/(1-x) --> B The Attempt at a Solution First of all, multiply (w-1) on both sides, then we get...
  13. K

    [Complex variables] z^4 - 4z^2 +4 -2i = 0

    Homework Statement "This is an example from my textbook: Solve the equation z4 - 4z2 + 4 - 2i = 0 Solution: Rearranging, we get z4 - 4z2 + 4 = 2i or (z2 - 2)2 = 2i = (1+i)2 This has solutions z2 - 2 = 1+i or -1-i. Equivalently z2=3+i or z2=1-i These may be solved to give the 4...
  14. M

    Complex variables, De moivres formula

    Homework Statement z is a complex number different from 1 and n >= 1 is an integer 1 + z + z^2+ ... + z^n = \frac{z^{n+1} - 1}{z-1} show that: \sin(\theta) + \sin(2 \theta)+ ... \sin(n \theta) = \frac{ \sin(n \theta/2) \sin((n+1) \theta / 2)}{\sin(\theta / 2)} The Attempt at a Solution...
  15. K

    Which Textbook Is Best for Learning Several Complex Variables?

    Can anyone recommend a good textbook for an undergrad who has done real and complex analysis and wants to learn about several complex variables? Thanks!
  16. pellman

    When derivates w/ resp to complex variables differ from real derivatives?

    If z is complex, the following rules are true, right? \frac{d}{dz}z^n = nz^{n-1} \frac{d}{dz}\ln{g(z)} = \frac{1}{g(z)} \frac{d}{dz}g(z) \frac{d}{dz}e^{g(z)}=e^{g(z)}\frac{d}{dz}g(z) These are of course the same rules as for real variables. When do I need to be careful about...
  17. E

    Complex variables conformal mapping trig identity

    Homework Statement map the function \begin{equation}w = \Big(\frac{z-1}{z+1}\Big)^{2} \end{equation} on some domain which contains z=e^{i\theta}. \theta between 0 and \pi Hint: Map the semicircular arc bounding the top of the disc by putting $z=e^{i\theta}$ in the above formula. The...
  18. P

    Graphing functions of two complex variables.

    Hello, I have come across this problem in my studies where I need to try to come up with a graph of a function involving two complex numbers. I have been trying to figure this out for a while now, but I am not sure how to do it. Is there any way to do this type of thing by hand or in Maple...
  19. A

    Functions of Complex Variables

    Homework Statement {Q 6.2.2 from Arfken "Mathematical Methods for Physicists"} Having shown that the real part u(x,y) and imaginary part v(x,y) of an analytic function w(z) each satisfy Laplace's equation, show that u(x,y) and v(x,y) cannot have either a maximum or a minimum in the interior of...
  20. S

    Evaluate integral by using residues: complex variables

    Homework Statement evaluate: \int_{-\infty}^{\infty}e^{-ax^2}cos(bx)dx Homework Equations The Attempt at a Solution since \cos(bx) is the real part of: e^(b*x*i), we can rewrite the integral as: Real[\int_{-\infty}^{\infty}e^{bxi-ax^2}dx] but now I am stuck because I...
  21. J

    Solving Complex Variables Homework

    Homework Statement I'd like some help with 2 problems: Show by using Demoivre's theorem and the geometric series formula that the sum of all n values of z^(1/n) is zero when n >=2. Z is a complex number. Use the geometric series formula and Demoivre's theorem to show that: Homework Equations...
  22. B

    Proofs for Complex Inequalities to Bound a Complex Integral

    Hi, I need two simple proofs of complex inequalities. 1) |1-z|/|z|<2 2)|1+z|/|z|>1/2 Ik need them for a bound of a complex integral.It's not homework Thank you
  23. W

    Exploring the Applications of Complex Variables in Physics and Engineering

    I just completed a course on complex variables. I really enjoyed the application sections. I was thinking of studying CV a little more on my own. The question is: is it worth it to study more CV for physics and engineering? What advantage would it give me? Aside: I was browsing through...
  24. C

    Complex Variables: Prove f is Constant

    Homework Statement Let z be a complex variable Suppose f is an entire function and Re(f(z))\leq c for all z Show that f is constant. (Hint: Consider exp(f(z)) Homework Equations possibly this: e^z=e^x(cos(y)+isin(y)) where z=x+iy The Attempt at a Solution I had no idea how I...
  25. S

    Complex Variables: Expressing f(z)=cos(z)

    Homework Statement let: f(z)=u(x,y)+iv(x,y) I want to express the following function like the one above: f(z)=cos(z)\equiv=\frac{1}{2}(e^{iz}+e^{-iz}) Homework Equations (i=sqrt(-1)) f(z) is a complex function The Attempt at a Solution...
  26. J

    Properties of the modulus - complex variables question

    Given f(z) = (z+1) / (z-1) for z not equal to 1 My teacher wrote |f(z)| = |x+1+iy| / |x-1+iy| = sqrt((x+1)^2 +1) / sqrt((x-1)^2 + 1) How do the values within the modulus work out to the right hand side? I can't figure it out.
  27. J

    How Does the Function f(z) = (z + 1) / (z - 1) Map Complex Regions?

    For z not equal to 1 f(z) = (z + 1) / (z - 1) How do you show the function maps {z ϵ C : Re(z) < 0} into {w ϵ C : |w| < 1} and {w ϵ C : |w| < 1} into {z ϵ C : Re(z) < 0}? ---- I don't even know how to start this one besides that "into" means 1-1. How do you show the mappings?
  28. W

    What is the nature of the singularity at z=0 for the function f(z)=1/cos(z)+1/z?

    Homework Statement Determine the nature of the singularity at z=0Homework Equations f(z)=\frac{1}{cos(z)}+\frac{1}{z} The Attempt at a Solution by expanding into series: f(z)=\Sigma_{n=0}^{\infty} \frac{(2n)! (-1)^n}{x^{2n}} + \Sigma_{n=0}^{\infty} (-1)^n (z-1)^n Now \frac{1}{z} has no...
  29. D

    Complex Variables Question (should be easy)

    Homework Statement Sketch the set of points determined \mid 2\bar z + i \mid = 4. Homework Equations \mid z - z_0 \mid = r and \mid \bar z \mid = \mid z \mid The Attempt at a Solution I know that it will be the circle with radius = 4 and z_0 is the center of the circle and that...
  30. C

    Complex Variables: Need help with Chebyshek poly and de Moivre Theorem

    Homework Statement The nth order Chebyshev polynomial is defined by Tn(x)= cos( n arccos(x) ) , n is a positive integer; -1<= x <= 1. Using the de Moivre theorem, show that Tn(x) has the polynomial representation Tn(x)= 1/2 [(x+sqrt(x2-1))n+(x-sqrt(x2-1))n] The Attempt at a Solution I really...
  31. E

    Textbook of complex variables function

    Could someone recommend some good textbooks of complex variables function. I am a physics student with poor mathematical knowledges, so I want to study some mathematical subjects again, by myself. thanks!
  32. S

    Proving Analytic Functions with Complex Variables

    Homework Statement Suppose f(z) is an analytic function on domain D, and suppose that, for all z in D, we have 2*Re(f(z)) + 3*Im(f(z))=12. prove that f(z) must be a constant. Homework Equations The Attempt at a Solution ok, I am drawing somewhat of blank with this one but I am...
  33. S

    Line Integrals (Complex Variables)

    Homework Statement i) Define a path \gamma whose image is the ellipse \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 traced counterclockwise. ii) Show that \int \frac{1}{z} dz = \int \frac{1}{z} dz for a suitable circle \beta (NOTE: THE FIRST INTEGRAL IS OVER THE ELLIPSE \gamma, THE SECOND ONE IS...
  34. S

    Complex variables: Logarithm function.

    I'm taking a complex variables course, and I'm really stuck at it, I've never felt this way in any math course before :S, I'm starting to get angry. Anyway here is the problem, I hope someone can give me a hand. I believe this is a basic and simple problem in the subject... Homework Statement...
  35. B

    The level of difficulty of complex variables vs. Real analysis

    Which course is more difficulty in terms of which subject contains more rigorous proofs, Complex variables or Real analysis. I don't know whether I should dropped Complex variables, but the only reason I am taken it is because of the useful physics applications found in this course. I my...
  36. B

    Complex variables : open connected sets

    Homework Statement Let S be the open set consisting of all points such that |z|<1 or |z-2|<1 . State why S is not connected. Homework Equations The Attempt at a Solution According to my complex variables book the definition of a connected set are pairs of points that can be...
  37. B

    Complex variables : Triangle inequality

    Homework Statement Using the fact that |z(1)-z(2)| is the distance between two points z(1) and z(2) , give a geometric argument that a)|a-4*i| + |z+4*i| =10 represents an ellipse whose foci are (0,4) and(0,-4). Homework Equations Triangle inequality equation; distance formula...
  38. L

    Gunning and Rossi's several complex variables

    Does their book have exercises? tanks Laura
  39. L

    Complex variables transformation/mapping of w=z^3

    Hi all! My basic problem is that I can't figure out how to do a transformation of z^3 from the w to the z plane. Homework Statement w=z^3. Region R in the w plane, R = {-1 \leq u \leq 0, 0\leqv\leq1 }. Region Q is the mapping of region R onto the z plane. Sketch region Q. The...
  40. M

    Complex Variables: Missing Step in Example

    Homework Statement Integral from 0 to 2*pi of [(Cos(3*theta)) / (5 - 4Cos(theta))] (d*theta) Homework Equations z=exp(i*theta), Cos(theta) = (z + z^(-1)) / 2, Cos(3*theta) = ((exp(3*i*theta)+exp(-3*i*theta)) / 2 = (z^(3) + z^(-3)) / 2, dz = (i*z) (d*theta) The Attempt...
  41. M

    Complex Variables integration formulas

    Homework Statement Let C. denote the circle |z-z.|= R, taken counter clockwise. use the parametric representation z= z. + Re^(io) (-pi </= o </= pi) for C. to derive the following integration formulas: integral C. (dz/(z-z.)) = 2ipi Homework Equations note: z. and C. represent z knot...
  42. S

    Proving Integral of Log(z+5) is 0 Around Contour z=1

    Can anyone help me with this pls? How can you prove that the integral of f(z) around the contour z= 1 is 0 where f(z) is Log(z+5) Thx I know Log(z) is ln r + i (theta). But i don't know how that applies to this situation. Also, do I solve it as a normal integral or use...
  43. B

    Complex variables, maximum value

    Homework Equations Let w \in C be a fixed complex number with |w|<1. Let f(z)=\frac{z-w}{1-\bar{w}z}. Calculate its maximum value in the region |z| \leq 1. The Attempt at a Solution How should I approach this? Not sure if maximum modulus principle is of much help.
  44. B

    Computing Integrals with Complex Analysis

    Homework Equations Using complex analysis, compute \int_{-\infty}^{\infty} \frac{e^{itx}}{1+x^2}dx where t is real. The Attempt at a Solution I'm not good at complex analysis at all and am totally lost. I do know some Fourier analysis though and using it I got \pi e^{-|t|}. How should I...
  45. B

    Are complex variables used in physics

    I'm thinking of taking an introduction to complex variables course as an elective for my applied math major(or minor). Are their any applications of complex variables in physics?
  46. nicksauce

    Analytic Functions and the Gauss Mean Value Theorem: Proving an Inequality

    Homework Statement Let f be analytic in the disk |z| <= 1. Prove that for any 0 < r < 1, |f(0)|^2 <= \frac{1}{\pi r^2} \int \int_{x^2 + y^2 <= r^2} |f(z)|^2 dxdy Homework Equations The hint is apply the Gauss mean value theorem on f^2(z)The Attempt at a Solution Having difficulty starting...
  47. E

    Complex Variables. Complex square root function

    Homework Statement Proof that f(z)=\sqrt{z}=e^{\frac{\ln z}{2}} with logarithm branch [0,2\pi). Then f maps horizontal and vertical lines in A=\mathbb{C}-\{\mathbb{R}^{+}\cup\{0\}\} on hyperbola branches. Homework Equations I have that \ln_{[0,2\pi)} (z)=\ln\vert z\vert+i\mathop{\rm...
  48. E

    Complex Variables. Problem about complex sine.

    [SOLVED] Complex Variables. Problem about complex sine. Homework Statement Proof that the function \begin{displaymath} \begin{array}{cccc} f: &A=\left\{z\in\mathbb{C}\mid-\frac{\pi}{2}<\Re z<\frac{\pi}{2}\right\} &\longrightarrow &B=\mathbb{C}-\left\{z\in\mathbb{C}\mid...
  49. nicksauce

    Interesting complex variables problem

    Homework Statement Let \Omega be a bounded domain in C whose boundary is a curve z = z(t), a<=t<=b, and let A(\Omega) be the area of \Omega. Prove that A(\Omega) = \frac{1}{2}\int^b_a |z(t)|^2 Im(\frac{z'(t)}{z(t)})dt Homework Equations The Attempt at a Solution Not even sure...
  50. nicksauce

    Equilateral Triangles and Complex Variables: Proving the Relationship

    Homework Statement Let {z1,z2,z3} be complex variables such that |z1| = |z2| = |z3|. Prove that z1,z2,z3 are vertices of an equilateral triangle iff z1 + z2 + z3 = 0. Homework Equations The Attempt at a Solution Not really sure where to start on this. I know that |z2-z1= |z3-z2|...
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