Complex function Definition and 137 Threads

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)...

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?

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...

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...

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...

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...

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...

$$ f:\mathbb{C}\rightarrow\mathbb{C} \\ f(z)=\left\{\begin{array} \frac{(\bar{z})^2}/ {z} \quad z\neq0 \\ 0 \quad z=0 \end{array} \right.$$ 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...

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...

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...

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...

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.

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.

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

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...

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...

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...

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...

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...

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...

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...

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?

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 +...

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...

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...

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...

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.

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...

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...

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?

$$ g(z) = z^{87} + 36z^{57} + 71z^{4} + z^3 - z + 1 $$ 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?

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).

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 ... ...

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...

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...

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

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...

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

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)...

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.

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...