What is Complex function: Definition and 140 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).
When a variable in ##[\text { } ]## means its principal value, ##(-\pi,\pi]##, which is correct:
##Log(z^2)=log([z]^2)## or ##Log(z^2)=log([z^2])## (both, neither)?
My answer: one branch point is ##1## of the order 1, another is ##i## of the order 2.
My question is, how can I be sure that these are the only branch points?
hi guys
i found this problem in a set of lecture notes I have in complex analysis, is the following function real:
$$
f(z)=\frac{1+z}{1-z}\;\;, z=x+iy
$$
simple enough we get
$$
f=\frac{1+x+iy}{1-x-iy}=
$$
after multiplying by the complex conjugate of the denominator and simplification
$$...
First of all I am not sure which type of singularity is ##z=0##?
\ln\frac{\sqrt{z^2+1}}{z}=\ln (1+\frac{1}{z^2})^{\frac{1}{2}}=\frac{1}{2}\ln (1+\frac{1}{z^2})=\frac{1}{2}\sum^{\infty}_{n=0}(-1)^{n}\frac{(\frac{1}{z^2})^{n+1}}{n+1}
It looks like that ##Res[f(z),z=0]=0##
Hi,
I'm trying to find the residue of $$f(z) = \frac{z^2}{(z^2 + a^2)^2}$$
Since I have 2 singularities which are double poles.
I'm using this formula
$$Res f(± ia) = \lim_{z\to\ \pm ia}(\frac{1}{(2-1)!} \frac{d}{dz}(\frac{(z \pm a)^2 z^2}{(z^2 + a^2)^2}) )$$
then,
$$\lim_{z\to\ \pm ia}...
Hey everyone! I got stuck with one of my homework questions. I don't 100% understand the question, let alone how I should get started with the problem.
The picture shows the whole problem, but I think I managed doing the a and b parts, just got stuck with c. How do I find the largest region in...
Hi,
I have to find the real and imaginary parts and then using Cauchy Riemann calculate ##\frac{df}{dz}##
First, ##\frac{df}{dz} = \frac{1}{(1+z)^2}##
Then, ##f(z)= \frac{1}{1+z} = \frac{1}{1+ x +iy} => \frac{1+x}{(1+x)^2 +y^2} - \frac{-iy}{(1+x^2) + y^2}##
thus, ##\frac{df}{dz} =...
I’m coming at this question with a physics application in mind so apologies if my language is a bit sloppy in places but I think the answer to my question is grounded in math so I’ll post it here.
Say I have a function F(z) defined in the complex z plane which has branch points at z=0 and z =...
Hello,
I have to prove that the complex valued function $$f(z) = Re\big(\frac{\cos z}{\exp{z}}\big) $$ is harmonic on the whole complex plane.
This exercice immediately follows a chapter on the extension of the usual functions (trigonometric and the exponential) to the complex plane, so I tend...
Below are plots of the function ##e^{0.25(x-3)^{-2}} - 0.87 e^{(x-3.5)^{-2}}##
The first plot is for real values. It has a minimum at the red dot. The second plot has in its argument the same real part as the red dot, but has the imaginary part changing from -0.3 to 0.3. It shows the resulting...
When I type in this:
D [
Re[
Exp[u + 10*I]
],
u
] /. u->0.5
I get this output:
Of course, I could just put the Re outside and the D inside, but it would be nice to know what is wrong with the above. What's with the Re' in the output?
## u_x = 3x^2 -3y^2 ## and ## v_y = -3y^2-3x^2 ##
## u_y = -6xy## and ## v_x = -6xy##
To be analytic a function must satisfy ##u_x = v_y## and ##u_y = -v_x##
Both these conditions are met by x=0 and y taking any value so I think the functions is analytic anywhere on the line x=0
However...
The singularities occur at ##z = \pm i\lambda##. As ##\frac{d}{dz}(z^2+\lambda^2)^2|_{z=\pm i\lambda}=0##, these singularities aren't first order and the residues cannot be calculated with differentiating the denominator and evaluating it at the singularities. What is the general method to...
Homework Statement
Hi
I am looking at this proof that , if on an open connected set, U,there exists a convergent sequence of on this open set, and f(z_n) is zero for any such n, for a holomorphic function, then f(z) is identically zero everywhere.
##f: u \to C##Please see attachment...
Hi. I would like to ask regarding this function that keeps on cropping up on my study (see picture below).
What I did is simply substitute values for A and b and I noticed that it ALWAYS results to a real number. If possible, I would like to obtain the "non imaginary" function that is...
This is not a homework problem, I just am confused a little about the differences between a Nyquist plot and the plot of a complex function. I believe they are the same given the domain of the plot of a complex function is for all real numbers equal to or greater than zero. However, I am having...
Consider the function $f(z) = e^{1/z}$,
Show that for any complex number ${w}_{0} \ne 0$ and any δ > 0, there exists ${z}_{0} ∈ C$
such that $ 0 < |{z}_{0}| < δ$ and $f({z}_{0}) = {w}_{0}$
I really don't know where to begin on this.
Homework Statement
Homework Equations
First find poles and then use residue theorem.
The Attempt at a Solution
Book answer is A. But there's no way I'm getting A. The 81 in numerator doesn't cancel off.
Homework Statement
Find the derivative of ##f(z)=\frac{1.5z+3i}{7.5iz-15}##
Homework EquationsThe Attempt at a Solution
I had no difficulty using the standard derivative formulas to find the derivative of this function, but the actual result, that the derivative is zero, is confusing. For real...
Homework Statement
Homework Equations
Using Cauchy Integration Formula
If function is analytic throughout the contour, then integraton = 0. If function is not analytic at point 'a' inside contour, then integration is 2*3.14*i* fn(a) divide by n!
f(a) is numerator.
The Attempt at a Solution...
Homework Statement
It's not a homework problem itself, but rather a general method that I imagine is similar to homework. For a given elementary complex function in the form of the product, sum or quotient of polynomials, there are conventional methods for converting them to polar form. The...
Homework Statement
Suppose z = x + iy. Where are the following functions diﬀerentiable? Where are they holomorphic? Which are entire?
the function is f(z) = e-xe-iy
Homework Equations
∂u/∂x = ∂v/∂y
∂u/∂y = -∂v/∂x
The Attempt at a Solution
f(z) = e-xe-iy
I convert it to polar form:
f(z) =...
Homework Statement
How do the values of the following functions move in the complex plane when t (a positive real number) goes to positive infinity?
y=t^2
y=1+i*t^2[/B]
y=(2+3*i)/t
The Attempt at a Solution
I thought:
y=t^2 - along a part of a line that does not pass through the...
Homework Statement
[/B]
Find and classify the isolated singularities of the following:
$$ f(z) = \frac {1}{e^z - 1}$$
Homework EquationsThe Attempt at a Solution
I have the solution for the positions of the singularities, which is: ## z = 2n\pi i## (for ##n = 0, \pm 1, \pm 2, ...##) and this...
I am trying to find the limit of ## \frac {z^2 + i}{z^4 - 1} ## as ## z ## approaches ##i##.
I've broken the solution down to: ##\frac {(z + \sqrt{i})(z - \sqrt{i})}{(z+1)(z-1)(z+i)(z-i)} ## but this does not seem to get me anywhere. The solution says ## -0.5 ## but I don't quite understand how...
Hey all, I need the complex version of the sigmoid function in standard form, that is to say $$f(\alpha) =\frac{1}{1+e^{-\alpha}} , \hspace{2mm}\alpha = a+bi , \hspace{2mm} \mathbb{C} \to \mathbb{C}$$ in the simplified form: $$f = m+ni$$ but found this challenging, for some reason i assumed...
For a harmonic function of a complex number ##z##, ##F(z)=\frac{1}{z}##, which can be put as ##F(z)=f(z)+g(\bar{z})##and satisfies ##\partial_xg=i\partial_yg##. But this function can also be put as ##F(z)=\frac{\bar{z}}{x^2+y^2}## which does not satisfy that derivative equation!
Sorry, I...
Hi,
I'm not sure about the the normal vector N on a complex function
z(x,t) = A e^{i(\omega t + \alpha x)}
My approach is that (\overline{z} beeing the conjugate of z):
\Re{(\mathbf{N})} = \frac{1}{\sqrt{\frac{1}{4}(\partial x + \overline{\partial x} )^2 + \frac{1}{4}(\partial z +...
Hello everyone,
I have a problem with finding a residue of a function:
f(z)={\frac{z^3*exp(1/z)}{(1+z)}} in infinity.
I tried to present it in Laurent series:
\frac{z^3}{1+z} sum_{n=0}^\infty\frac{1}{n!z^n}
I know that residue will be equal to coefficient a_{-1}, but i don't know how to find it.
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...
Homework Statement
Let ##D={z : |z| <1}##. How many zeros (counted according to multiplicty) does the function ##f(z)=2z^4-2z^3+2z^2-2z+9## have in ##D##? Prove that you answer is correct.
Homework Equations 3. The Attempt at a Solution [/B]
The function has no zeros in ##D##, which can be...
#Hi All,
Let ## f: \mathbb C \rightarrow \mathbb C ## be entire, i.e., analytic in the whole Complex plane. By one of Picard's theorems, ##f ## must be onto , except possibly for one value, called the lacunary value.
Question: say ##0## is the lacunary value of ##f ##. Must ## f ## be of the...
what is the nature of singularity of the function f(x)=exp(-1/z) where z is a complex number?
now i arrive at two different results by progressing in two different ways.
1) if we expand the series f(z)=1-1/z+1/2!(z^2)-... then i can say that z=0 is an essential singularity.
2) now again if i...
Homework Statement
Evaluate the integral using any method:
∫C (z10) / (z - (1/2))(z10 + 2), where C : |z| = 1
Homework Equations
∫C f(z) dz = 2πi*(Σki=1 Resp_i f(z)
The Attempt at a Solution
Rewrote the function as (1/(z-(1/2)))*(1/(1+(2/z^10))). Not sure if Laurent series expansion is the...
Homework Statement
I'm struggling with the proof that the limit of a complex function is unique. I'm struggling to see how |L-f(z*)| + |f(z*) - l'| < ε + ε is obtained.
Homework Equations
0 < |z-z0| < δ implies |f(z) - L| < ε, where L is the limit of f(z) as z→z0 .The Attempt at a Solution...
Hi PF!
I am trying to run the following plot:
k = .001;
figure;
hold on
[X,Y]=meshgrid(-4:0.01:4);
a = 5.56*10^14;
b = .15/(2*.143*10^(-6));
for n = 1:8
k = k*2^(n-1);
Z = a./(X.^2+Y.^2).*exp(b.*(X-sqrt(X.^2+Y.^2)))-k;
contour3(X,Y,Z)
end
which works great if a = b = 1. But now...
Hi,
I need to plot the last function of this:
But I don't know how to generate the sum. I know the for loop is totally wrong, but I can't go any further. This is what I have:
Can someone fix the summation loop part for me?
Thanks in advance
Up to this point I have got a grasp of some basics of "steepest descent method" to evaluate the integral of a complex exponential function ##f(z) = \exp(A(x,y))\exp(iB(x,y))##. Using this method the original integration path is modified in such a way that it passes through its saddle points...
Homework Statement
Find the maximum value of f(z) = exp(z) over | z - (1 + i) | ≤ 1
Homework Equations
|f(z)| yields the maximum value
The Attempt at a Solution
f(z) = exp(x) ( cosy + i siny)
Unfortunately that's all I've got. I've seen examples with polynomials, but not with trigonometric...
Hi,
I am currently teaching myself complex analysis (using Stein and Shakarchi) and wondered if someone can guide me with this:
Find all the complex numbers z∈ C such that f(z)=z cos (z ̅).
[z ̅ is z-bar, the complex conjugate).
Thanks!
Homework Statement
##z(t) = t + it^2## and ##f(z) = z^2 = (x^2 - y^2) + 2iyx##
Homework EquationsThe Attempt at a Solution
Because ##f(z)## is analytic everywhere in the plane, the integral of ##f(z)## between the points ##z(1) = (1,1)## and ##z(3) = (3,9)## is independent of the contour (the...
Homework Statement
Hello, I have been tasked with the next problem, I have to prove that the next two integrals are complex numbers; but I have no idea of how to attack this problem.
Homework Equations
∫dx f*(x) x (-ih) (∂/∂x) f(x) integrating between -∞ and ∞
∫dx f*(x) (-ih) (∂/∂x) (x f(x))...
Homework Statement
What are the region of validity of the following?
1/[z2(z3+2)] = 1/z3 - 1/(6z) +4/z10
Homework EquationsThe Attempt at a Solution
Knowing that this is the expansion around z=0, I am trying to find the singularities of the complex function.
Which is when z2(z3+2) = 0
I...
Homework Statement
We are given that ##f(z) = u(x,y) + iv(x,y)## and that the function is differentiable at the point ##z_0 = x_0 + iy_0##. We are asked to determine the directional derivative of ##f##
1. along the line ##x=x_0##, and
2. along the line ##y=y_0##.
in terms of ##u## and...
The problem is to determine whether the function
##f(z) =
\left\{\begin{array}{l}
\frac{\overline{z}^2}{z}~~~if~~~z \ne 0 \\
0 ~~~if~~~z=0
\end{array}\right.##
is differentiable at the point ##z=0##. My two initial thoughts were to show that the function was not continuous at the point...
Hello everyone,
How do I find the limit of a complex function from the definition of a limit? For instance, consider the limit
##lim_{z \rightarrow -3} (5z+4i)##.
Would I simply conjecture that ##5z + 4i## approaches ##5(-3) + 4i## as ##z \rightarrow -3##; and then use the definition of a...
I've been told that the method scipy.optimize.Newton() will solve complex functions so long as the first derivative is provided. I can't make it work. The documentation for Newton() mentions nothing of complex functions. Could someone show me how one would find the roots of a function like f(z)...
I have this characteristic equation for the wave number eigenvalues k_n of a homogeneous infinite cylinder of radius R:
D_{m} = (k_n R) = 0,
where
D_m (z) = n_r J'_m(n_r z)H_m(z) - J_m(n_r z)H'_m(z)
and n_r is the refractive index of the cylinder, the bessel and hankel functions are...
Homework Statement
Calculate ##\int _Kz^2exp(\frac{2}{z})dz## where ##K## is unit circle.Homework Equations
The Attempt at a Solution
Hmmm, I am having some troubles here. Here is how I tried:
In general ##\int _\gamma f(z)dz=2\pi i\sum_{k=1}^{n}I(\gamma,a_k)Res(f,a_k)## where in my case...