Complex function Definition and 140 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. C

    Construct a complex function with these properties

    Homework Statement Construct a function f:C \rightarrow C such that f(x+y)=f(x)+f(y) and f(xy)=f(x)f(y) (aside from the identity function) Hint: i^2=-1 what are the possible values of f(i). The Attempt at a Solution All I've been able to do so far is come up with some (hopefully correct)...
  2. S

    MHB Differentiability of complex function

    I have found a question Prove that f(z)=Re(z) is not differentiable at any point. According to me f(z)=Re(z)=Re(x+iy)=x which is differentiable everywhere. Then where is the mistake?
  3. F

    Check if the complex function is differentiable

    The question is to check where the following complex function is differentiable. w=z \left| z\right| w=\sqrt{x^2+y^2} (x+i y) u = x\sqrt{x^2+y^2} v = y\sqrt{x^2+y^2} Using the Cauchy Riemann equations \frac{\partial }{\partial x}u=\frac{\partial }{\partial y}v...
  4. D

    Laurent expansion for a complex function with 3 singularites

    Homework Statement Hey guys, So I need a bit of help with this question: Find three Laurent expansions around the origin, valid in three regions you should specify, for the function f(z)=\frac{30}{(1+z)(z-2)(3+z)} Homework Equations None that I know of...just binomial expansion...
  5. D

    Laurent expansion for a complex function

    Homework Statement Expand f(z)=\frac{1}{z-4} in a laurent series valid for (a) |z|<4 and (b) |z|>4 Homework Equations The formula for laurent expansion... \sum_{n=-∞}^{+∞}a_{n}(z-z_{0})^{n} where a_{n}=\frac{1}{2\pi i}\oint_c \frac{f(z)}{(z-z_{0})^{n+1}}dz The Attempt at a...
  6. J

    Taylor series for a complex function

    Homework Statement Find the 5 jet of the following function at z=0: f(z) = \frac{sinhz}{1+exp(z^3)} Homework Equations \frac{1}{1-z}=\sum_{n=0}^\infty z^n where z=-exp(z^3) The Attempt at a Solution I have tried to multiply the series for sinhz by the series for \frac{1}{1-(-exp(z^3))} but...
  7. G

    Fourier series of complex function

    Homework Statement Hello guys, I have problem with the Fourier series, since we had only one lecture about it and I cannot find anything similar to my problem in internet. should we consider for the first f(x+1) integrated from -1 to 0 ? http://img819.imageshack.us/img819/3508/wbve.jpg when...
  8. S

    MHB Differentiability of a Complex Function

    f:\mathbb{C}\rightarrow\mathbb{C} \\ f(z)=\left\{\begin{array} \frac{(\bar{z})^2}/ {z} \quad z\neq0 \\ 0 \quad z=0 \end{array} \right. Show that f is differentiable at z=0, but the Cauchy Riemann Equations hold at z=0. Well i have tried to start the first part but i am stuck, could you...
  9. H

    Finding Amplitude and Phase of Complex Functions

    I can't seem to remember how to find the amplitude/phase of a complex function (I do know what to do for complex numbers, though). I know it's in my mind somewhere, but I just can't remember lol. So, for example, how would I find the amplitude/phase of: 3+j5t and 3ej4t EDIT: I know for...
  10. J

    Antiderive complex function f(z) and express as power series

    Let F(z) be the anti-derivative of the function f(z) = cos(z^3) with F(0) = 0. Express F(z) as a power series around z=0, giving both the first 3 non-zero terms and the general (nth) term. Hey guys really struggling with this integration and how to then express this as a power series. Any...
  11. D

    How do I integrate this continuous complex function?

    1. Homework Statement [/b] \int _{C} Re z^{2} dz clock wise around the boundary of a square that has vertices of 0, i, 1+i, 1.Homework Equations \int_{c} f(z) dz = \int \stackrel{b}{ _{a}} f[z(t)] \stackrel{\cdot}{z(t)}dtThe Attempt at a Solution Since it is piece-wise continuous I know I need...
  12. E

    Solving Complex Function for multiple solutions

    1. Find all the solutions to the equation z^4 + j^4 = 0 2. z^n = |a|e^j(Θ + 2pik) 3. I really don't know where to start, I thought about j^4 = 1, so z^4=-1. I then simplified to conclude that z^4 = -e^jpi. I am not sure if that is correct and if it is what to do next.
  13. B

    Find where a complex function is differentiable

    Using the definition of the derivative find at which points the function f(z) = Im(z)/z conjugate is complex differentiable. I know that it is not complex differentiable anywhere but I need to show it using the definition and not the Cauchy Riemann equations.
  14. S

    TI 89 Help complex Function Evaluation

    Is it possible to evaluate the limit of a complex function on a ti 89?
  15. P

    Why Does My Calculation of h(z) = Re(z) / Im(z) Yield -12i Instead of 12?

    Hi, so for a homework problem I have to evaluate these complex functions. The one I am having trouble on is: evaluate h(z) = Re(z) / Im(z) where z = (5-2i) / (2 - i) The answer is in the back of the book, which says that the solution is 12, however I keep getting -12i for my answer. I...
  16. R

    Continuity of a complex function defined on the union of an open and closed set

    Homework Statement (i) Let U and V be open subsets of C with a function f defined on U \cup V suppose that both restrictions, f_u \mathrm{and} f_v are continuous. Show that f is continuous. (ii) Illustrate by a specific example that this may not hold if one of the sets U, V is not open...
  17. R

    How many branches does a complex function have?

    Homework Statement How many branches does the function f(z) = \sqrt{z(1-z)} have on the set \Omega = \mathbb{C} \backslash [0,1] Homework Equations The Attempt at a Solution Not really sure how to go about it at all. Our lecturer didn't say too much about branches but...
  18. M

    Line Integral of a complex function

    I'm trying to solve this integral as x-> Infinity \int \frac{dz}{8i + z^2} ...on the y=x line, but I have no idea what I'm doing. The book I'm using is less than helpful in this regard. I'm not supposed to use any complex analysis tools (Cauchy, etc), but just solve it as a line...
  19. P

    Expressing a complex function as polar coordinates

    Homework Statement Consider the complex function f (z) = (1 + i)^z with z ε ℂ. 1. Express f in polar coordinates. Homework Equations The main derived equations are in the following section, there is no 'special rule' that I (to my knowledge) need to apply here. The Attempt at a...
  20. L

    What is the Image of the Complex Function f(z)=z+1/z on |z|>1?

    Let f(z)=z+\frac{1}{z}, the question is to find the image of this function on |z|>1. To do so, I tried to find the image of the unit circle which is the interval [-2,2] and so I could not determine our image. If also we tried to find the image of f we get f(re^{i\theta})=u+iv where...
  21. L

    Find limit of complex function

    Homework Statement Let z = x + iy and let f(z) = 3xy + i(x - y2). Find limz→3 + 2i f(z). Homework Equations The definition of a limit. The Attempt at a Solution I did f(3 + 2i) = 18 - i It seems pretty clear that it is a continuous function, but I can't prove it. So I tried using the...
  22. J

    Whats a way to determine the range of this complex function?

    Describe the range of p(z) = -2z^3 for z in the quarter disk |z| < 1, 0 <Argz < \frac{\pi}{2}. The answer is the circular sector |w| < -2, -\pi <Argw < \frac{\pi}{2} What's a good way of seeing why this is true?
  23. M

    Transfer Function: Magnitude and Phase of Complex Function

    Homework Statement f(s) = f(\sigma + j\omega) = \frac{1}{(1+s)^2} Find the magnitude and phase angle of f(j\omega) Homework Equations s = j\omega is a substitution you can make, but I'm not sure if you are supposed to apply that here The Attempt at a Solution I tried substituting \sigma +...
  24. C

    Maximum Value Complex Function

    Homework Statement Find the maximum value of |(z-1)(z+1/2)| for |z|≤1. Homework Equations Calculus min/max concepts? The Attempt at a Solution Let f(z)=|(z-1)(z+1/2)|. Observe f(z) is the product of 2 analytic functions on |z|≤1, g(z)=z-1 and h(z)=z+1/2. Therefore f(z) is analytic on...
  25. A

    How Is a Complex Function Expanded in an Annular Region?

    I have an exercise that says the following: Expand the following function as a series: f(z) = \frac{1}{z-1} + \frac{1}{2-z} for 1<lzl<2 The result is attached, but I don't really understand what has been done. Therefore tell me: How is that series generated? Initially I thought I...
  26. X

    Exam problem essiential singulairity or not complex function

    Homework Statement 1/(1-cos(z)) Homework Equations taylor expansion at 0 for cos(z)=1-x^2/2+x^4/24 and so on. 1/(x^2/2+x^4/24...) The Attempt at a Solution Because all of the powers are negative wouldn't that make it a essential singularity and 0 + 2∏n. Also it just explodes at...
  27. T

    Partial fraction decomposition with complex function

    As part of a project I have been working on I fin it necessary to manipulate the following expression. e^(icx)/(x^2 + a^2)^2 for a,c > 0 I would like to decomp it into the form A/(x^2 + a^2) + B/(x^2 + a^2) = e^(icx)/(x^2 + a^2)^2 but I am having trouble getting a usable outcome.
  28. L

    Finding Symmetric Poles for Complex Function Integrals

    I'm looking for a function which has two simple poles, and whose integral along the positive real axis from 0 to infinity is equal to its integral along the positive imaginary axis. I don't really know where to start. I'm looking at functions which have symmetry with respect to real/imaginary...
  29. A

    Calculating the residue of a complex function

    Homework Statement calculate the residue of the pole at z=i of the function f(z)=(1+z^2)^-3 State the order of the pole Homework Equations I know the residue theorem and also the laurent series expansion but I'm having trouble applying these The Attempt at a Solution I...
  30. C

    A problem in limit of a complex function

    Homework Statement I'm a newbie at complex analysis. Find: lim \frac{\bar{z}^{2}}{z} z→0 2. The attempt at a solution L'Hospital rule gives the answer in no time. But how do you solve without it?
  31. D

    MHB Analyzing the Complex Function $g(z)$

    $$ g(z) = z^{87} + 36z^{57} + 71z^{4} + z^3 - z + 1 $$ For $|z|<1$. Let $f(z) = 71z^4$. Then $|f(z) - g(z)| = |-z^{87} - 36z^{57} - z^3 + z - 1| \leq |z|^{87} + 36|z|^{57} + |z|^3 + |z| + 1 < 71|z^4|$ So g has the same number of zeros as f which is 0 with multiplicity of 4. Correct?
  32. H

    What is the general form of an analytic function with ux = 0 on a given domain?

    In the question below I do not understand what is meant by the "general form". 'Suppose f(z) = u(x,y) + i*v(x,y) is analytic on a domain D, and ux = 0 on D. Find the general form of f(z).
  33. G

    Complex function of several variables

    Hi! While studying Global Cauchy Theorems in complex analysis, I've realized that I need to know a definition of continuity of a complex function of several variables... Thus, I ask you the definition of continuity of a complex function of several complex variables. What I mean is ... ...
  34. G

    Derivative of a complex function in terms of real and imaginary parts.

    Hi, I wonder if anyone knows when (maybe always?) it is true that, where z=x+iy \text{ and } f : \mathbb{C} \to \mathbb{C} \text{ is expressed as } f=u+iv, \text{ that } f'(z)=\frac{\partial u}{\partial x}+i\frac{\partial v}{\partial x}? I'm pretty sure that this is true for f=exp. I...
  35. O

    Derivative of a complex function wrt x_i

    Given a function \sum_{j=0}^m(b_{ij}-euclid(x_i, x_j))^2, where euclid(x_i, x_j) denotes the Euclidean distance (1D or 2D) between x_i and x_j. I'm supposed to find the derivative with respect to x_i. The sum sign and the dimensionality are the problem for me. Any help on how to solve...
  36. P

    Is f:y=|ln(x)| a Complex Function?

    hi, I'd like to ask whether the function f:y=|ln(x)| (|| denotes the absolute value) is complex. I'm a little bit confused because it is defined also for negative numbers in reals. Is the "negative part" classified as a complex function even if it has no imaginary part ? thanx
  37. J

    Integrating a Complex Function: A Challenge

    Homework Statement Integrate the following: (sin(x)/x)^4 between negative infinity and infinity. Homework Equations The residue theorem, contour integral techniques. The answer should be 2pi/3 The Attempt at a Solution I'm not even sure where to start honestly. I define a function...
  38. M

    How do you visualize the complex function (1+i)i and its multivalued nature?

    How do you plot (1+i)i, where i is the imaginary number. I decomposed it to eilog√2e-∏/4e2∏n (n = 0, +1, +2, ...) Should it be some kind of lattice? I would imagine it's discontinuous due to the n Thanks
  39. I

    How do I find the magnitude of a complex function?

    Homework Statement I'm asked to find the magnitude of a complex function R(jw) = 1 + exp(-jw) + exp(-j2w) + exp(-j3w) + exp(-j4w) R(jω) = 1 + \exp{(-jω)} + \exp{(-j2ω)} + \exp{(-j3ω)} + \exp{(-j4ω)} where ω is the angular frequency j is the imaginary number j = \sqrt{-1} and \exp(-jnw)...
  40. C

    Find f(z) when |z|<=1 and |f(z)|=|e^z| for |z|=1

    f(z) is analytic and not equal to zero in the unit circle (|z|<=1) . we also know that |f(z)|=|e^z| for |z|=1. find f(z) How should i approch to this question? I know that on inside the boundry it can't get the maximum nor the minimum .. but it doesn't help at all. i have no idea what to do.
  41. A

    MATLAB Drawing Complex Function Graph in MATLAB

    Dear fellows I need to draw a graph of a complex function in matlab, but could not do so, can you please help me in this issue?? I have mentioned each and every constant required for function(complex function), y1=1.9417+3.5012i; y2=1.9417+3.5012i; y3=4.7348-1.4837; y4=4.7345+1.4837...
  42. B

    Residue of a complex function with essential singularity

    Homework Statement Hello friends, I'm a student of mechanical engineering and I have a problem with computing residues of a complex function. I've read some useful comments. Now I ve got some ideas about essential singularity and series expansion in computing the residue. However, I still...
  43. H

    Analyzing Complex Function z^a: Derivative & Analytic Region

    Homework Statement Define z^a = exp(a log z), assume a is complex Where is this function analytic, and what is its derivative. Homework Equations Log z is defined as log z = ln |z| + i*arg z, 0 <= arg z < 2pi. The Attempt at a Solution I am really unsure of how to look at this...
  44. TheFerruccio

    Plotting the Image of a Complex Function: w=1/z

    Homework Statement Find/sketch the image of the function under the transform w = 1/z Homework Equations x=-1 The Attempt at a Solution So, I decided to take the mapping 1/z as 1/(x+iy) For x=-1: \begin{align} w=\frac{1}{z}&=&\frac{1}{x+iy}&=&\frac{-1-iy}{1+y^2} \end{align} Getting u in...
  45. R

    Line integral of complex function

    I have to evaluate this line integral in the complex plane by direct integration, not using Cauchy's integral theorems, although if I see if a theorem applies, I can use it to check. \int (z^2 - z) dz between i + 1 and 0 a) along the line y=x b) along the broken line x=0 from 0 to 1...
  46. K

    Cube Roots of a Complex Function Am I doing something wrong?

    1. Find the cube roots of the complex number 8+8i and plot them on an Argand diagram Thats the problem, I've had a go at the solution and came up with 3 solutions using the \sqrt[n]{r}*(cos(\frac{\theta+2\pi*k}{n})+isin(\frac{\theta+2\pi*k}{n})), but the answers (roots) I get, I can't plot it...
  47. TheFerruccio

    Finding the complex function given a real component

    Homework Statement Find f(z) = u(x,y) + iv(x,y) with u or v as a given.Homework Equations u = \frac{x}{x^2+y^2}The Attempt at a Solution Using the Cauchy-Riemann equations, if the function is analytic, then u_x = v_y and u_y = -v_x So, the first thing I did was find the x derivative of u...
  48. M

    Derivative of Complex Function

    Hi guys, I was hoping someone can check my work finding the following complex derivative: Homework Statement Find where the function is differentiable / holomorphic: f(z) = e^-x * e^-iy Homework Equations I know I must satisfy the Cauchy - Reimann equations. The Attempt at a...
  49. M

    Taking the limit of a complex function

    Homework Statement Hi guys. I was hoping you could help me find the limit of a complex function. So here goes: The lim z --> i of [i(z)^3 - 1 ] / (z+i) The Attempt at a Solution If z approaches i, then (x,y) approaches (0,1) Do I let z = x+iy, then expand out the cube and plug...
  50. E

    Least-square optimization of a complex function

    Dear all, I have a least square optimization problem stated as below \xi(z_1, z_2) = \sum_{i=1}^{M} ||r(z_1, z_2)||^2 where \xi denotes the cost function and r denotes the residual and is a complex function of z_1, z_2. My question is around ||\cdot||. Many textbooks only deal with...
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