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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
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How do you plot complex analysis graphs on the Geogebra platform?
How do you find books, information, and articles about it?
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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.
Thank you all
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Is the Poincaré Conjecture relatively easy to prove for the case of simply connected real analytic 3- manifolds ?
Stiefel Manifolds and Grassmann manifolds are two examples of Smooth/analytic Riemannian Manifolds, with its far reaching applications in data analysis, such as classification, clustering and object tracking etc.
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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.
Me also I am interested in your topic and it is very important issue
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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 ?
For a function describing some physical property, when complex arguments and complex results are physically meaningful, then often the physics requires the function to be analytic. But if the only physically valid arguments and results are real values, then the physics only requires (infinitely) smooth functions.
For example, exp(-1/z^2) is not analytic at z=0, but exp(-1/x^2) is infinitely smooth everywhere on the real line (and so may be valid physically).
One place where this happens is in using centre manifolds to rigorously construct low-D model of high-D dynamical systems. One may start with an analytic high-D system (e.g., dx/dt=-xy, dy/dt=-y+x^2) and find that the (slow) centre manifold typically is only locally infinitely smooth described by the divergent series (e.g., y=x^2+2x^4+12x^4+112x^6+1360x^8+... from section 4.5.2 in http://bookstore.siam.org/mm20/). Other examples show a low-D centre manifold model is often only finitely smooth in some finite domain, again despite the analyticity of the original system.
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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)
Dr Spiros Konstantogiannis excellent solution
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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?
Seyed Jamaleddin Mostafavi Yazdi sir, did you get what you were doing wrong?
I am also doing a similar analysis and stuck at the same problem.
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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?
A conformal map is a function in mathematics that preserves angles but not necessarily lengths locally. In more technical terms, let and be open subsets of At a location, a function is said to be conformal (or angle-preserving) if it retains angles between directed curves while also retaining orientation.
the conformal map, A transformation of one graph into another in which the angle of intersection of any two lines or curves remains constant. The most prominent example is the Mercator map, which is a two-dimensional depiction of the earth's surface that includes compass directions.
In contrast, if the Cauchy-Riemann equations are met and the derivative at the point is not zero, one may demonstrate that there is an a > 0 and such that the above is true. As a result, the map retains angles. As a result, a map is a conformal map if and only if it is a one-to-one, onto analytic function of D to D.
Kind Regards
Qamar Ul Islam
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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.
In your case the NBO analysis is useless. As you can see from the output in the NBO analysis part, NBO analysis believes that Cu has 5 lone pairs of electrons, while its bonding with oxygens is fully ignored. This is a common problem of NBO, the bond with low covalency is usually not exhibited by BD type of NBO orbital.
I suggest you use Multiwfn (http://sobereva.com/multiwfn) to perform atom-in-molecules (AIM) analysis to characterize the Cu-O bonds, you can easily find large number of publications using AIM theory to study bonds in transition-metal coordinates.
There are many other methods in Multiwfn could be used to study the bonding, including various kinds of bond orders, ETA-NOCV, bond order density (BOD), charge displacement analysis (CDA) and so on, please check "Multiwfn quick start.pdf", which can be found in Multiwfn package.
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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 …
Are you sure you have defined your function correctly?
1. Usually z=x+iy. But in your function z is in the limit, thus being in both the arguments and what the integral is computed against. If z is not x+iy, the function is not a function of (x,y).
2. What do you mean by limit? Do you want to compute the case when z->0?
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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
Hi In what area was the issue raised? Euclidean space, Hilbert space, Banach space?
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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?
Hello Saima,
The "right" answer depends on your specific research question(s), and the way in which the outcome variable is quantified. If the DV is genuinely metric (interval strength and continuous), then you might consider:
A 2 x 2 x 2 ancova, in which the pretest score is used as the covariate, the two factors are group orientation and sex of participant, and the third IV is a repeated measures factor (score after intervention 1, score after intervention 2). This design allows for a number of questions to be addressed. However, given the small sample/cell size, the statistical tests are not going to be very powerful unless the magnitude of an effect is quite substantial.
The same data may be equivalently analyzed via a regression model. The same admonition about statistical power will apply.
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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
Carlos, There is a wealth of information about the Riemann Zeta function. It started out as the Dirichlet series which converged and was holomorphic for abs(z)>1. However, through the process of analytic continuation the Zeta function could be defined as a meromorphic function on the entire complex plane, holomorphic on C-{1} with a simple pole at z=1.
Often times when one extends a convergent series representation to homomorphically to a larger region which is done to generate the Zeta function one finds the function is no longer single valued.
For example when considers the simple sqrt(z), then runs into the problem that it is multivalued and hence cannot be extended homomorphically to C. This gave rise to the concept of a Riemann surface, covering spaces, branch points and branch cut so to address this issue which can often arise in analytic continuation. http://www1.spms.ntu.edu.sg/~ydchong/teaching/07_branch_cuts.pdf
The Zeta function when extended to the p-adic number field, however is not single valued and has branch points.
There are many books at all levels on the zeta function. It is one of the most important special functions of mathematics and foundational to analytic number theory. Just go an Amazon and search on Riemann Zeta function in Books and see what pops up.
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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.
Keep the original coordinates of the DNA-ligand complex from docking. Define all the necessary force field parameters.
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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!
R software provide better and meaningful results.
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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?
If anyone proposes a theory and no one can make it wrong, at least it is not wrong, but it remains a mystery, and that is why it is called a mystery .. This does not mean that the theory is wrong.
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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?
I agree with you. Although year after year we are getting closer. A few year ago, it was proven that at least 40% of zeros have to be on the critical line. So a lot of progress has been done. Many conjecture have also been presented, which if proven, they would imply the Riemann Hypothesis. So have gotten some important result, however, we have not quite solved it yet.
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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?
Hi all. Here is a more or less elementary solution. Since the function extends to a holomorphic one on the complement of the half line ]-\infty,-1] (of course, we are using the standard determination of log), we can use a contour which is slightly on the right of -1. See the attached file.
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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 working with both, In my view, Adams is very simpler and user friendly. Modeling, defining contact, defining belt, gears, chain and other mechanical part in Adams is very simple and user friendly related to using SimScape.
The advantage of SimScape is that you can use it simply with Simulink for example for designing a controller for a system, while Adams should coupled with Simulink.
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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.
I just typeset my answer for you, and I am glad that it looked like it was taken from a book, maybe it will be from the book I will write in the future.
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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?
In mathematics, Cauchy's integral formula, named after Augustin-Louis Cauchy, is a central statement in complex analysis. It expresses the fact that a holomorphic function defined on a disk is completely determined by its values on the boundary of the disk, and it provides integral formulas for all derivatives of a holomorphic function.
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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.
Dear Dr. Feng Qi ,
The further, the more you will be convinced of the correctness of my words. Maybe. I wrote these words for the reason that I have passed a similar path here in August and September this year:
Certain personalities promised to destroy my proof - to show that my proof was wrong. Then they forgot it, apparently? I made a reminder 5 days ago. But, no one reacted and insignificant discussions were continued.
My answer is this: waiting is always the best tactic and strategy. Remember the RF's mathematician G. Perelman. This person generally did not give a damn about the opinion of the professional community of mathematicians and was absolutely indifferent to any action that disturbed him. This is a real Hero!
Best Regards,
Sergey
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Basic knowledge of Complex Analysis and Nevanlinna value distribution theory.
Dear Anil,
In connection to your question, if e^z+1/z is the derivative of a function, which omits the small function 1/z, then what do you think about the parent function f, which has Picard exceptional value(s) ?
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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.
The real or the imaginary part of any complex function is harmonic. The mapping is to transform a question in complex plane domain A (not easy to solve) to B (easy to solve or even known). Then you transform the solution back to your original domain.
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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?.
Is the Schwarz lemma also relevant in matrix domains? Let's say the first type is in the classical domains? The first type of matrix ball?
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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
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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.
You may use multiple kernel or multiple PCA, also you may see some good incitement from this old paper :
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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?
In addition to Sandhya Avasthi's post, here is a connex feed on RS:
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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?
In a paper that can be found on arXiv or at , LeClair gives a reasonably accurate algorithm to estimate the non-trivial zeros up to 10^200=Googol^2. My paper that can be found at Cogent Mathematics, on arxiv or RG
gives an estimate that bounds the n'th zero and checks LeClairs result for the number Googol. Although both these are not iterative, and work only for non-trivial zeros that sit on the critical line, they are predictive and easily calculated. Once a zero is estimated, or bounded, it's accurate value can then be found from formula given.
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Is there any transformation from fuzzy to the real or complex domain??
See the following references may are interesting
(1) [IEEE 7th Seminar on Neural Network Applications in Electrical Engineering, 2004. NEUREL 2004. 2004 - Belgrade, Serbia and Montenegro (Sept. 23-25, 2004)] 7th Seminar on Neural Network Applications in Electrical Engineering, 2004. NEUREL 2004. 2004 - From fuzzy to real sets
(2) [IEEE [1993] Second IEEE International Conference on Fuzzy Systems - San Francisco, CA, USA (28 March-1 April 1993)] [Proceedings 1993] Second IEEE International Conference on Fuzzy Systems - From fuzzy logic to fuzzy truth-valued logic for expert systems: a survey
de Mantaras, R.L., Godo, L.
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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)
See the following references
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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?
The introduction of new technologies/treatments is always problematic. Where there are clinical (and cost) risks, there is rightly a degree of conservatism. The evidence required to overcome conservatism and professional inertia, scales with risk, benefits and costs. This is something that we have discussed in Campbell & Knox (2016) International Journal of Technology Assessment in Health Care, 32(3), 122-125. doi:10.1017/S0266462316000234.
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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...
All the best
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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.
Dear Ahmed
According to me by the comment, the reviewer wanted to know how your proposed fuzzy multi-objective model is beneficial over the existing or what is the advantages of you proposed model. If your paper is an working paper then why you have particularly chosen or preferred the method over others.
Wish you the very best.
With regards
Syed Abou Iltaf Hussain
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complexity analysis tools
Thank you Jack Son
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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.
Dear Boris,
For holomorphic function we have the maximum-modulus principle:
A theorem expressing one of the basic properties of the modulus of an analytic function. Let f(z) be a regular analytic, or holomorphic, function of n complex variables z=(z_1,…,z_n), n=1, defined on an (open) domain D of the complex space C^n into C^m, which is not a constant. The local formulation of the maximum-modulus principle asserts that the modulus of f(z) does not have a local maximum at a point z_0 in D.
Some researchers define closed submanifold as compact without boundary (perhaps, you may clarify the defition). With this definition I do not see that there is any explicit representation of complex ball as closed submanifold.
best,MM
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Some news about state of the art about Steve Smale 21th 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). Gianluca
According to Wikipedia, it is still open in full generality.
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if 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.
Dear
Thanks for your collaboration, and suggestions.
-Shatha
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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?
It has been quite long since the original question was posed, but if anybody is still interested, I think the answer is in the first formula of Section A.1.2 in this paper
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Is the series in the picture convergent? If it is convergent, what is the sum of the series?
Dear Joachim,
you've right. Just ignore the first part of my answer. May be the double series calculation will lead to some result.
Rgrds,
Tibor
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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?
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Analytical Solution
This stress in Normal Stress in X direction and it is induced stress. Diagram is a Complex plane showing Real X axis and Imaginary Y axis. Although magnitude of imaginary part is very less but on plotting on complex plane it is showing this interesting feature but i am not getting what this behavior signifies.
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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.  xn + yn = zn;     2.  x(n + ε)/2 = zn/2  + yn/2;   and  3.  x(n - ε)/2 = zn/2 - yn/2 where
e = xn/2,  f = xε/2,  a = zn/2, and b = yn/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', 3rd Edition, Serge Lang, 2005, Springer,  ISBN 0-387-22025-9.  Please refer to relevant pages, 171 - 176,  of textbook.
_________________________________________________________________
'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’,
'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,
'GEOMETRY OF NUMBERS WITH APPLICATIONS TO NUMBER THEORY',
Dear David
You can see Emzari Papava's page on ABC theorem
Sincerely Yuri
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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.
Did you make sure that your sample is fully saturated at 9000 Oe? If your sample is not fully saturated, you may have only measured a minor cycle. Minor cycles can be shifted in any directions, depending on the history of the sample.
Does your VSM have a superconducting magnet? Another thing that can explain your shift may be the presence of residual magnetic fields in your magnet. Some magnetic flux sometimes get trapped in superconducting magnets and can polarize your sample (the zero point is biased). Usually the value and the direction of that residual field depend on the magnetic history of the magnet.
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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
Dear Colleague
Nevanlinna theory is part (most) of value distribution theory of holomorphic functions There are many books where you can find abaout Nevannlina theory.. I recomend  you text where you can lfind about Nevannlina Theory, which you can find; on the following web sit:  https://math.berkeley.edu/~vojta/cime/cimebeamer.pdf
Best regards and many success,
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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.
No, this is a mistake. I do not have probability generating function when discussing heavy tails. May be it is mentioned in the text of my talk, but this talk is partially connected to the problem under discussion. Of course, I may use statistical estimator for probability generating function (and/or for characteristic function), but it is not enough to obtain asymptotic behavior of the tail.
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What is the best or simplest reference, which says that a bounded subharmonic function can be extended over a condimension 2 subset?
Thank you for your answers. My question should be more precisely as following:
If a continuous function f on a compact manifold M is superharmonic (with the given Riemannian metrics) on M-N with N a subset of codimension 2. Does it means that
f is constant?
This seemly was a general fact as early as 1981. In the paper:
"Y. T. Siu & P. Yang: Compact K¨ahler-Einstein surfaces of nonpositive bisectional curvature, Invent. Math. 64 (1981), 471–487."
it was mentioned as a fact without even mentioning a reference.
In my recent paper:
"On Bisectional Nonpositively Curved Compact K¨ahler Einstein Surfaces"
to appear in Pacific J. of Math.
Anyway, for our special case, my earlier post reduced the problem to the
harmonic case.
A similar argument proved the result we need for the special case as following:
"Therefore, ∆f = 0 on M−N. A similar arguments shows that \int _hd (f−g)dτ = 0, where g is the f value of the corresponding point on hs for any given s < d. Let s tends to zero, we get \int _hd fdτ = 0. By f ≥ 0 on hd we obtained that f = 0 near N. Therefore f extends over N as a harmonic function. This implies that f = 0 on M."
The last sentence came from the Hodge Theorem.
This solved the special case in our paper.
One more question is following:
Does this proof work, in some more general sense, for the general case as I mentioned
at the beginning of this post?
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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?
Let us consider two dimensional problems, where the power of complex analysis can be seen quite directly.
If a function f(x,y)=u(x,y)+ i v(x,y) is differentiable at z0=x0 + i y0 then at this point ux=vy and uy = -vx is satisfied. These are Cauchy-Riemann conditions. This immediately has a important consequence for quantum theory. It means that the quantum mechanical wave function (which has to be differentiable)  is analytic everywhere.
When z=x+i y and f(x,y)=u(x,y)+ i v(x,y), then if f(x,y) is an analytic function it immediately implies that u(x,y) and v(x,y) should satisfy Laplace's equation. This is directly related with Physics because then both real part and complex part of an analytic function (such as the wave function) must be harmonic. Example: The free particle wave function e i k.r .
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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)?
.
My answer from above gives a sufficient condition (on g) for the desired conclusion on f, and it is independent on any differentiability hypothesis. It follows by simple handling inequalities. Hence this implication holds and one can say something on this subject.
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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.
Dear friend,
I think MATLAB software can be a good choice. A helpful link.
Good Luck.
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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.
The formula (1.2) for completely monotonic function on an interval is not valid. For example, f(t) = e-t - e-1 is completely monotonic on [0, 1] with f(1) = 0, and the latter property is impossible for a non-zero function satisfying the integral representation (1.2). In order to have (1.2) you need a function that has completely monotonic extension on the semi-axes (0, \infty). There is really a lot of information about absolutely monotonic / completely monotonic functions in the original Bernstein's paper
S. N. Bernstein (1928). "Sur les fonctions absolument monotones". Acta Mathematica 52: 1–66. doi:10.1007/BF02592679
Maybe some of this information can be of use for you. S. N. Bernstein is one of the most famous mathematicians from my University, and it is always a pleasure for me to mention his works: they contain much more material than the textbooks citing his results.
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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
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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?
In higher codimension, we cannot define the regular infinite one type with  tangent vector field and the levy form directly like in C^2. But it is in some sense possible : see my paper with J.F Barraud  (at least for define finite type):
Barraud, Jean-François; Mazzilli, Emmanuel
Regular type of real hyper-surfaces in (almost) complex manifolds. (English) Zbl 1082.32017
Math. Z. 248, No. 4, 757-772 (2004). This is done for regular one type. Now for the D'angelo type : singular type, it is possible see the paper :
Barraud, Jean-François; Mazzilli, Emmanuel
Lie brackets and singular type of real hypersurfaces. (English) Zbl 1154.32009
Math. Z. 261, No. 1, 143-147 (2009). I do not know if it is exactly a think like you want but perhaps it is helpfull for you. regards E/.Mazzilli.
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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?
I think your condition wants to be
X g(Y,Z) = g(\nabla_X Y , Z) + g(Y, \nabla_X Z)
just as it is for the Levi Civita connection. Equivalently \nabla g = 0.
Also I think you mean a Norden Metric.
For a Levi Civita we have for the torsion T = 0. For a Norden metric on a almost complex manifold people choose metrics that also have \nabla J = 0, and have a condition on teh torsion.
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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*}
Before computing the limit one has to fix the precise meaning of the expression. The functions $g(z) = z^s$ and $h(z) = \ln(z)$ are multi-valued, so the question will be correctly posed only after selecting appropriate branches of all expressions that  include these functions. Outside of this I see no difficulties.
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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.
Hi colleague Feng Qi
I just had the intention to help you, but the Professor Yao Liang Chung was faster than me.
Best regards and wishes,
Mirjana Vukovic
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Is there any applications of conformal mapping in fluid dynamics or precisely in the field of bubble dynamics?
Sure, but you need the system to be quasi-2d, e.g. a Hele-Shaw cell. Check out the recent paper by Giovanni Vasconcelos:
"Multiple bubbles and fingers in a Hele-Shaw channel: complete set of steady solutions"
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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.
Have a look at the following papers:
E.O Tuck, On positivity of Fourier transforms.
Bull. Austral. Math. Soc., 74(1), 133-138, 2006.
S. Albeverio and C. Cebulla, Müntz formula and zero free regions for the Riemann
zeta function. Bull. Sci. Math., 131 (2007) 12-38.
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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).
Please dear Peter explain more this state:
That means that kx != n mod 2n for each power x from 0 to n-1.
Best
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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?
Even though the critical mass is rapidly increasing (seminars, conferences, papers, books all around the world) it is clear that complexity science remains still as alternative thinking. It is far from being "normal science". No problem with that.
The truth is that normal science has a clear intention to coopt complexity science, but it can't. It has already coopted, say, complex thinking (= Morin), system science and some close fields. I personally find very troublesome that many complexologists do see non-lieanrity but they work on to via analytical methods. Pum! They kill thus non-linearity.
As Katherine says, education is a feasible and necessary road/tool. Yet, I am not convinced that it is the only one.
Let me please put it in the following terms: vis-à-vis the systemic and systematic crises that going on out there, we need a more radical mind set. That is certainly not system thinking. It is complex science - at large. Hence, a set of actions and plans are to be undertaken, I guess.
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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?
In  metal-complex formation reactions, the thiourea act as ligand and forms coordination bond with Copper. The reduction depends on the strength between the metal and sulfur... It is a stable complex.
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The equation is given in the attached picture. ai and bi are known complex numbers, and λ is the complex unknown. I would like to estimate the solution which has the largest modulus.
On a first glance,
from 1=\sum a_i/(b_i-\lambda) we conclude that
there is k such that n|a_k|\geq |b_k-\lambda|.
Hence  \lambda   is in the disk   B(b_k; n|a_k| )  and therefore
(1)  |\lambda|\leq |b_k| + n|a_k|.
1.  For example, if  for i\neq k,  a_i=0,
then   \lambda=b_k -a_k.
2. For  example,  a_i=b_i=1,  1\leq I \leq n, then  \lambda=1-n.
Is this rough estimate for your purposes ?
3. if  $b_i= -s a_i$, s\geq 0,  and  |b_i| + n|a_i|= c,  for  1\leq I \leq n,     then  |\lambda|=c . Thus we can have equality in (1).
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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.
Va multumesc mult domnule Profesor.
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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.
Regards
Edoardo
Dear Wiwat,
My apologies for the delayed response. Thank you for sharing this material with me, it is enlightening, to say the least.  In spite of my time issues I am very inclined to give it a try and program it myself. Depending of how long it takes me I either chose your way or the more pragmatical approach and use SVDPACK library.
Thank you all, for all the useful information that you provided me with.
Best regards
Edoardo
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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.
Dear  Khaled Mohamed,
Suppose S is a compact metric surface, then  for each point p, there
exits a local coordinate chart (U,f ), such that p belongs U and the
local coordinates are isothermal  ( and metric conformal in local coordinates ) . In particular,   it is true for surfaces  in Euclidean  3-dimensional space.
Ahlfors, Lars V. (1966), Lectures on quasiconformal mappings, Van Nostrand
So using  Beltrami  equation, we can   local coordinates  in which metric is conformal.
best,
MM
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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.
A very good book is the following;
Complex Analysis for Mathematics and Engineering
Third Edition
John H. Mathews
Russell W. Howell
1997 by Jones and Bartlett Publishers
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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?
Yes, the Cauchy-Riemann condition is true for all analytic real valued functions. In which case they can be used to prove that the class of analytic real valued functions consists of the real constants.
But the Cauchy-Riemann conditions does not hold for arbitrary functions of two variables, regardless of they are real-, imaginary-, or complex-valued.
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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?
Well, I would recommend not to take holism as if there was symmetry - or asymmetry between the components, here the public and the private sector. Properly speaking, a good complexity approach does not focus so much in holism as in networks, fluctuations or non-linear dynamics. Systems theory is much more concerned with holism, as such.
The interactions between the private and the public sector depend on each country. In one cases the private sector is larger and stronger. In some others, quite the opposite happens. Still, in some other cases, there is a sort of evenness between both sectors.
Furthermore, I would suggest to take always into account the weight of the so-called third sector, i.e. the civil society. You cannot not discharge it - here or there.
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