Science topics: AnalysisComplex Analysis

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# Complex Analysis - Science topic

Explore the latest questions and answers in Complex Analysis, and find Complex Analysis experts.

Questions related to Complex Analysis

How do you plot complex analysis graphs on the Geogebra platform?

How do you find books, information, and articles about it?

The attached image is the introduction of

**part D - Complex Analysis**from [Kreyszig, E. (2009). Advanced engineering mathematics, 10th edition.]. At the end of the introduction, the author explains why complex analysis is important in three points. I am having difficulty understanding**point (3)**, my questions are: 1- How studying analytic functions as functions of a complex variable leads to a deeper understanding of their properties? 2- How studying analytic functions as functions of a complex variable leads to interrelations in complex that have no analog in real calculus. I am grateful to anyone who could or tried to answer my questions and if possible provide**examples**to get a full understanding.Is the Poincaré Conjecture relatively easy to prove for the case of simply connected real analytic 3- manifolds ?

I want to know the complexity analysis (Time and Space ) of the algorithm of multi-task learning deep neural network-based compared to the conventional algorithms.

Consider the powerful central role of differential equations in physics and applied mathematics.

In the theory of ordinary differential equations and in dynamical systems we generally consider smooth or C^k class solutions. In partial differential equations we consider far more general solutions, involving distributions and Sobolev spaces.

I was wondering, what are the best examples or arguments that show that restriction to the analytic case is insufficient ?

What if we only consider ODEs with analytic coeficients and only consider analytic solutions. And likewise for PDEs. Here by "analytic" I mean real maps which can be extended to holomorphic ones. How would this affect the practical use of differential equations in physics and science in general ? Is there an example of a differential equation arising in physics (excluding quantum theory !) which only has C^k or smooth solutions and no analytic ones ?

It seems we could not even have an analytic version of the theory of distributions as there could be no test functions .There are no non-zero analytic functions with compact support.

Is Newtonian physics analytic ? Is Newton's law of gravitation only the first term in a Laurent expansion ? Can we add terms to obtain a better fit to experimental data without going relativistic ?

Maybe we can consider that the smooth category is used as a convenient approximation to the analytic category. The smooth category allows perfect locality. For instance, we can consider that a gravitational field dies off outside a finite radius.

Cosmologists usually consider space-time to be a manifold (although with possible "singularities"). Why a manifold rather than the adequate smooth analogue of an analytic space ?

Space = regular points, Matter and Energy = singular points ?

Hi, in a paper, I found the equation.

f(x)=(x+a-ib)^-1 - (x-a-ib)^-1

And, I'm going to get the answer.

integral[ Im[f(x)] ] = ?

(integrate from 0 to infinity)

I am analyzing the squeal of simple brake using Abaqus and I tried to find the unstable frequencies using complex frequency which caused the squealing noise; however, all of the modes have nearly zero dampings (lie on the imaginary
axis) and the pairs of modes which become coupled and formed a stable/unstable pair couldn't be found. I attach the brake model in the comment. How can I found some unstable frequencies using Abaqus?

Conformal mapping is a powerful tool to solve 2D boundary value problems that would would otherwise not be able to be solved analytically

Have these ideas been extended to 3D applications? Can someone point me to a suitable reference?

Hello,

I would like to prove that the type of coordination bond between copper and oxygen is dsp2 in the following complex structure. So I used NBO analysis, but there seems to be no related information. I wonder if there is anything else that needs to be applied.

I will receive all the advice with thanks. I also attached a file that used NBO analysis (.log).

Thank you.

*This paper is a project to build a new function. I will propose a form of this function and I let people help me to develop the idea of this project, and in the same time we will try to applied this function in other sciences as quantum mechanics, probability, electronics …*

Hello

Can someone help me to solve this?

Because I really don't know about these problems and still can't solve it until now

But I am still curious about the solutions

Hopefully you can make all the solutions

Sincerely

Wesley

We have 2 groups (total 8 participants, 4 in each group with 2 male and 2 female) of participants (one group on one end of the continuum and the other on the other end) with a score on a variable (Continuum). After applying intervention 1, we applied intervention 2 on both groups. Are we going to use paired samples t-test separately for both groups thrice (Once with pre-test score and intervention 1 and second time with pre-test score and intervention 2 and third time with intervention 1 and intervention 2)? How are we going to analyse this experimental data?

How to work properly in the development of an integral like the Abel Plana defined on this image:

I am interested in to have a set of steps for attacking the problem of developing the integral and to determine a criterion of convergence for any complex value s, I mean, when the integral could have some specifical behavior at, for example, s=1/2 + i t where I am interested in to study it.

I am interested in the proper evaluation of that integral only with formal steps into complex analysis.

The Abel PLana formula appears also on https://en.wikipedia.org/wiki/Abel%E2%80%93Plana_formula

Best regards

Carlos López

I have docked my dna and ligand and then save the best-docked structure in the form of pdb then I take the coordinates of my ligand and form the .itp and .gro file using PRODRG software. It gives me topol.top file then I add coordinates of ligand (.gro) file in DNA.gro file and increase the total no. of atoms and also added ligand.itp in topol.top and also added ligand 1 in molecules in topol.top even after this my solv.gro file of DNA and ligand does not show any interaction. i am not able to form ions.tpr file. i have attached solv.gro file.

Hi,

I'm a new user of MatLab and I want an objective way to identify how complicated an image is according to it's color variation. I can't use entropy() because it's for grayscale image. How can I do that?

Thanks!

Dear fellow researchers,

In your expert opinion, do you think the Riemann Hypothesis is true? The first billion zeros have been computed and they all verify this hypothesis. However, we have previously seen patterns hold until a very large number than break (There is a conjecture that holds for n<10^40). Do you think there is any reason to believe that it might be false?

How many more years do you predict it will take before the Riemann Hypothesis is solved?

Do you think we are close or does it seem that we are still very far?

Can anyone help me prove using Contour Integration that the value of integral log(1+z)/z from limit -1 to +1 is pi^2/4?

I would like to know the advantages in terms of complexity in the analysis (which software is easier to use). Also, I want to know wich software is more reliable for the analysis of mechanism.

I am attempting to solve the limit which involves the Reimann Function zeta

Lim [(zeta(1-z)zeta(1-zc))/(zeta(z)zeta(zc))]=1

as z tends to get closer to a root of the Reimann Function Zeta written here as zeta.

Here zc in the equation means the conjugate of the complex number z.

As we know, the Cauchy integral formula in the theory of complex functions holds for a complex function which is analytic on a simply connected domain and continuous on its boundary. This formula appears in many textbooks on complex functions.

My question is: where can I find a generalization of the Cauchy integral formula for a complex function which is analytic on a multiply connected domain and continuous on its boundary?

How to prove or where to find the integral inequality (3.3) involving the Laplace transforms, as showed in the pictures here? Is the integral inequality (3.3) valid for some general function $h(t)$ which is increasing and non-negative on the right semi-axis? Is the integral inequality (3.3) a special case of some general inequality? I have proved that the special function $h(t)$ has some properties in Lemma 2.2, but I haven't prove the integral inequality (3.3) yet. Wish you help me prove (3.3) for the special function $h(t)$ in Lemma 2.2 in the pictures.

Basic knowledge of Complex Analysis and Nevanlinna value distribution theory.

Conformal Mapping is used to solve 2D electrostatic phenomena, but what are the steps to follow, and how one get to know about which mapping function should use to map?

Is there any web reference or Book?

I want to solve Poisson's Equation in 2D geometry, please provide some good reference or Book for this.

Thank You in advance...

See for example the paper arXiv:1001.1805v1 [math.CV] 12 Jan 2010,

The Schwarz Lemma at the

Boundary by Steven G. Krantz. In this paper the author explores versions of the Schwarz lemma

at a boundary point of a domain (not just the disc). Estimates on

derivatives of the function, and other types of estimates as well, are considered.The author reviews recent results of several authors, and present some new theorems as well

It seems that this question is related to Q1. Is there a version of Jack's lemma for hyperbolic domains in space?.

I'm working on a project and I wanted to know the Time and Space complexity Analysis of MOEAs. I have searched the google scholar but there isn't any valuable information on this matter. please help me and let me know if there is a good comparative study or article on this subject.

thanks

Consider I am using multiple KDE bandwidths for my data and I would like to find out what the 'best-fit-model' would be. Is there anything like AIC for glm that I could use to accomplish this? I have used AUC and complexity analysis outlined in papers like Silva et al. 2018, but my Reviewer 2 has notified me that AUC isn't sufficient for this. After running the analysis on several nonsensical smoothing values I tend to agree.

Any assistance is appreciated.

Decision Trees are extremely fast in classification. However, they are slow in construction during learning phase. Is there any paper on complexity analysis of Multiway Split, Multi-Class Decision tree?

Compute nontrivial zeros of Riemann zeta function is an algebraically complex task. However, if someone able to prove such an iterative formula can be used to get all approximate nontrivial using an iterative formula, then its value is limitless.How ever to prove such an iterative formula is kind of a huge challenge. If somebody can proved such a formula what kind of impact will produce to Riemann hypothesis? . Also accuracy of approximately calculated non trivial accept as close calculation to non trivial zeros ?

Here I have been calculated and attached first 50 of approximate nontrivial using an iterative such formula that I have been proved. Also it is also can be produce millions of none trivial zeros. But I am very much voirie about its appearance of its accuracy !!. Are these calculations Is ok?

Can anyone please tell me about the complexity analysis in terms of Big oh of various image denoising techniques? Or I want to know how much time image denoising techniques are required for its operation...(Big oh notation)

Whenewer someone say that TMS, or complexity analysis or machine learning is 'novel methodology' I wander how long simething could be perceived as novel. In case if TMS even more than 30 years seems not to be enough. What do you think, what are the factors importannt in this issue?

Dear Professor, I have read your paper and it is very interesting to be used in the deformation analysis. But for this complex analysis, is there any supporting documents that can help us understand clearly on how he formula may work to the data until the output?

Thank you Sir

Time Series Analysis of 3D Coordinates Using Nonstochastic O...

I have a paper accepted recently subject to some corrections but not sure how to address one of the comments says ": " I think the authors should add a content about the complexity analysis of the fuzzy multi-objective model ". You help would be much appreciated.

More general, I want to get an algebra of holomorphic fuctions on n-dimesional complex ball as qoutient of algebra of holomorphic fuctions on C^k. I know there exists Remmert-Bishop-Narasimhan-Wiegmann embedding theorem, but it does not provide us any explicit representation of complex ball as closed submanifold.

Some news about state of the art about Steve Smale 21

^{th}century problems; in particular, I'm interested on "Mean Value Problem" (not inserted in original Smale's 1998 list, but formulated by Smale already in 1981, see Addenda in attached paper). Gianlucaif u be a harmonic function in sobolev space W^(1,p) (Omega) with zero trace (i.e; T(u)=0) and {Laplace operator} (u) =0, then u identically zero.

The quadratic term of a Taylor expansion of a multivariate scalar valued function can be expressed in terms of the Hessian. Is there a similar form for vector valued functions, in the sense that all partial derivatives can be arranged in a matrix or tensor?

Is the series in the picture convergent? If it is convergent, what is the sum of the series?

For example, we have a sequence,

S={-1,-1,1,1,0,-1,-1,-1,-1,-1,-1,-1,0,0,1,-1,1,1,0,-1,0,1,-1,1,-1,-1,1,1,0,-1,0,-1,0,0,-1,-1,0,-1,1,0,1,1,-1,0,0,1,-1,1}

Can we use, Berlekamp–Massey algorithm to find the linear complexity of the given sequence?

Thanks in advance.

**Since there has been considerable work done to answer this famous and important question, we expect a clear and definitive proof of the ABC Conjecture within 48 months.**

**_________________________________________________________________**

**On simple forms, we shall also consider the following mathematical statements:**

**e * f = a + b and e / f = a - b.**

**Example: We can consider the following Fermat's Equations (1, 2, and 3):**

**1. x**^{n}+ y^{n }= z^{n}; 2. x^{(n + ε)/2}= z^{n/2}+ y^{n/2}; and 3. x^{(n - ε)/2}= z^{n/2}- y^{n/2}where

**e = x**^{n/2}, f = x^{ε/2}, a = z^{n/2}, and b = y^{n/2}with the control parameter, ε, such that

**0 < ε < n.**

**_________________________________________________________________****"The ABC Conjecture: Given λ > 0 there exists a positive number, β(λ) > 0, which has the following property. For any nonzero relatively prime integers, a, b, and c such that**

**a + b = c,**

**we have:**

**max( |a|, |b|, |c| ) ≤ β(λ) * rad(a*b*c)**

^{1+λ}.**Notes:**

**We have relatively prime integers, a, b, and c or**

**gcd( a, b ) = gcd( a, c ) = gcd( b, c ) = 1.**

**Moreover, we define the radical, rad(), of some integer, k, such that | k | > 1 to be**

**rad(k) = ∏ p such that p | k for which rad(k) is the product of all primes, p, dividing k, taken with multiplicity one."**

**Book Reference: 'UNDERGRADUATE ALGEBRA', 3**

^{rd}Edition, Serge Lang, 2005, Springer, ISBN 0-387-22025-9. Please refer to relevant pages, 171 - 176, of textbook.

**_________________________________________________________________****Reference links:**

**'Easy as ABC? Not quite!',**

**'The probabilistic heuristic justification of the ABC conjecture',**

**'On Fermat's Equations',**

**‘DECOUPLING, EXPONENTIAL SUMS AND THE RIEMANN ZETA FUNCTION’,**

**"Proof of the Riemann Hypothesis", https://www.math10.com/forum/viewtopic.php?f=63&t=1549#p7077;**

**'ABC Conjecture and Riemann Hypothesis', http://rgmia.org/papers/v7n4/abc.pdf**

**'On a combination between an equivalence to the Riemann's Hypothesis and the abc conjecture',**

**'The Riemann Hypothesis: Arithmetic and Geometry ',**

**'From ABC to XYZ, or Addition versus Multiplication',**

**'SMOOTH SOLUTIONS TO THE abc EQUATION: THE xyz CONJECTURE',**

**'On the distribution in short intervals of integers having no large prime factor'**

**by Prof. J. C. L. et al,**

**'On Dyadic Models',**

**'GEOMETRY OF NUMBERS WITH APPLICATIONS TO NUMBER THEORY',**

In Magnetic measurements., VSM parameter results are given following with Ms = 2.5110 x10-3 emu/g ; Mr = -4.0825 x10-3 emu/g and Hc = 477.34 G. Can anyone import your advice.

Hi,

I'm wondering about any Applications of Theory of Nevanlinna in Other Disciplines, such as Physics.

Any Papers, Theses and books are important to me.

Best regards

I think, the old Markushevich's book on Analytic Functions is a good source of techniques and results in general. However, many particular cases may not fit those frames and may require specific analysis.

Best regards, Boyan.

What is the best or simplest reference, which says that a bounded subharmonic function can be extended over a condimension 2 subset?

Hi,

In Statistical signal processing, lot of research is based on complex analysis. Many techniques and methods are transformed to complex domain. Whereas complex information is only important in form of magnitude and phase. So whats the difference in using magnitude information or real and imaginary information of the data? Why is phase important? What the difference of the signal that is added with phase information and without phase information?

Appreciate your comments

Let f(x)+g(x) = h(x). Here, h(x) is minimum at the points(a1,a2,...,ak). For which condition , we can say that f(x) is also minimum at the points(a1,a2,...,ak)?

Thanks in advance for your idea and please give any reference

.

I am interested in using conformal maps for image distortion and I am looking for software environments that can do this. So far I came up with:

1) SeamlessMaker, a commercial stand alone program for image filtering etc., that has some conformal maps for selection: http://www.hypatiasoft.fr/

2) ComplexMapStill.jar, a free stand alone Java implementation by Christian Mercat with some predefined maps and the possibility to define new ones: https://karczmarczuk.users.greyc.fr/TEACH/InfoGeo/Work/ComplexMapStill.jar

3) ImageForwardTransformation[] in Mathematica: http://community.wolfram.com/groups/-/m/t/854405?p_p_auth=TLh6kMXL

Are there other implementations

1) in other programming languages: C/C++, Python, ...

2) in commercial Math environments: Maple, MathLab, ...

3) in free Math environments: Scilab, Octave, ...

For my art processes I would ideally need a command line program that takes a path to an input image, a description of a conformal map and a path for the resulting output image so that this could be called from my own scripts. But a batch process within a Math environment would also be useful.

What are completely monotonic functions on an interval $I$? See the picture 1.png

What is the Bernstein-Widder theorem for completely monotonic functions on the infinite interval $I=(0,\infty)$? See the picture 1.png

My question is: is there an anology on the finite interval $I=(a,b)$ of the Bernstein-Widder theorem for completely monotonic functions on the infinite interval $I=(0,\infty)$? In other words, if $f(x)$ is a completely monotonic function on the finite interval $I=(a,b)$, is there an integral representation like (1.2) in the picture 1.png for the completely monotonic function $f(x)$ on the finite interval $(a,b)$?

The answer to this question is very important for me. Anyway, thank everybody who would provide answers and who would pay attention on this question.

if i have a model like this;

proc logistic data=name;

where disease in (0,1);

class sex race presents education wealth region / param=ref;

model disease (event='1') = age sex race presents education wealth region age*education wealth*region region*education;

run;

How do I get the odds ratio for each of these;

age

sex

race

presents

education

wealth

region

age*education

wealth*region

region*education

using the oddratio statement in proc logistic? Since the variable region is invoved in two different interaction.

thanks

In two-dimensional case, we can follow Kohn's definition of type by using holomorphic tangent vector field and the Levi function to define infinite type(cite H. Kang, Holomorphic automorphisms of certain class of domains of infinite type, Tohoku Math. J. (2) 46 (1994), 435–442. MR 95f:32041). But when we consider in higher dimensions, can we still use this method to define infinite type?

g(\nabla_X Y, Z)= - g(Y, \nabla_X Z)

if this condition satisfies in Riemannian manifold

so can we take same condtion in Nordan metric?

if we can not take this condition so

what will be possible condition in Nordan -metric?

please give me valuable suggestions

How to compute the limit of a complex function below? Thanks.

For $b>a>0$, $x\in(-\infty,-a)$, $r>0$, $s\in\mathbb{R}$, and $i=\sqrt{-1}\,$, let

\begin{equation*}

f_{a,b;s}(x+ir)=

\begin{cases}

\ln\dfrac{(x+ir+b)^s-(x+ir+a)^s}s, & s\ne0;\\

\ln\ln\dfrac{x+ir+b}{x+ir+a}, & s=0.

\end{cases}

\end{equation*}

Compute the limit

\begin{equation*}

\lim_{r\to0^+}f_{a,b;s}(x+ir).

\end{equation*}

How to define and compute the power-exponential function of a complex variable? For example, how to define and compute the complex function $(\ln z)^{\sin z}$? where $z=x+iy$ is a complex variable. Thank you for your help.

Is there any applications of conformal mapping in fluid dynamics or precisely in the field of bubble dynamics?

One year ago I asked the same question and never found asnwer. So I do it again:

My purpose is to show that this integral

$$I_t(x)=\int_{-\infty}^{\infty}e^{-\frac{\cosh^2(u)}{2x}}\,e^{-\frac{u^2}{2 t}}\,\cos\left(\frac{\pi\,u}{2t }\right)\,\cosh(u)\,du,\hspace{0.5cm}x,t>0$$

is positive ( $I_t(x)>0$ for all $x,t>0$) . Believe me that I've tried with different forms of Riemann integral but I have not had success.

So, maybe I should try to consider the function as such (not as a

sum of areas) and look for something with the derivatives with respect to x or even with respect to t.

Please give me a hand. Any help is welcome.

I have tried to solve the problem, first I have proved that the cardinality of the following set $\{z\in \mathbb{C} : z^n=1 and z^k \neq -1, 0<k<n \} $ is $\phi(n)$ (Euler phi function).

After several decades of exponential growth of studies on complexity, it seems to me that society remains on the sidelines, as if all research and serious advances were considered a mere academic debate far from people's real needs. What should the complexity research community do to permeate society?

I was doing an experiment of Tris(thiourea)copper(I) sulfate synthesis (using the thiourea and Copper(II) sulfate pentahydrate) where thiourea was reduction agent. My question is, do compounds which contain C=S often act as a reduction agent and if they do what is the reason of such behaviour?

Is this case linked to PEARSON'S HARD & SOFT ACIDS & BASES (HSAB) THEORY?

The equation is given in the attached picture.

*a*and_{i}*b*are known complex numbers, and_{i}*λ*is the complex unknown. I would like to estimate the solution which has the largest modulus.This is just a observation on my side.

Bubble dynamics research has many difficulties due to nonlinearity of the system of differential equations, so can we overcome on this difficulties by complex analysis techniques?

I wait for any contribution, paper, book, ... etc.

Best Regards.

Hello,

My objective is to calculate the condition number of a sparse matrix in order to get an idea of how ill-conditioned it may be. The definition of the condition number does not give any help c.n(A) = ||A^-1|| ||A|| since I would have to calculate the inverse of the matrix. I found another posible way of calculate the c.n(A)= sqrt(largest eigenvalue of (A^T)(A)), hence my question.

Thanks in advance.

Regards

Edoardo

Dear colleagues:

Is there a conformal mapping that transforms non-spherical surfaces to spherical ones? or what is the methodology to construct that mapping?

if there is such mapping it's better to be convertible one.

Best Regards.

What is you suggestion about a text book on complex analysis for engineers that preparing them in solving application problems in circuits.. etc. I want this book to teach from it.

Cauchy-Riemann equations give the conditions that for which complex functions are differentiable in a complex mathematical space.

each real-valued function could be imagined as a complex function of imaginary part equal to zero, so the same conditions should be satisfied for them too; but it seems that these equations have no sense according to real-valued function.

the question is that should these conditions be satisfied for real-valued functions? if yes, how? ; if not, why?

Holism has its value in complexity of analysis, not dividing and not being reductionist. But how would You explain public and private sectors and the interaction between them by holistic approach?

Hi,

For a linear complex differential equation f^(k)+A_(k-1) f^(k-1)+...+A_0 f=0, the coefficients A_j are of finite (p,q)-order, I need to know the conditions on the coefficients that make all solutions of the equation above of infinite (p,q)-order.

I need papers, of any references .

Best regards