Complex analysis is the field of mathematics dealing with the study of complex numbers and functions of a complex variable. we notice is that the behaviour of $f$ near the essential singular In the $\,\frac{\infty}{\infty},$ case it also works*but*some care must be put in case we have that not just the variable $\,z\to \infty\,$ but only $\,Re(z)\to\infty\,\,\,or\,\,\,Im(z)\to\infty\,$ , as this can have some rather funny effects on some particular functions. Evaluate $\lim\limits_{z\to 0}f(z)$ and $\lim\limits_{z\to 2}f(z)$. In this case, when the value $f(0)=1/2$ is assigned, $f$ becomes entire. Example: Let's consider the examples above. A coordinate singularity occurs when an apparent singularity or discontinuity occurs in one coordinate frame, which can be removed by choosing a different frame. ) x Poles are one kind of singularity. Addition, multiplication, modulus, inverse. Removable singular point. (a) Find all z such that the power series converges. In addition to covering the basics of single variable calculus, the book outlines the mathematical method--the ability to express oneself with absolute precision and then to use logical proofs to establish that certain statements are Residues serve to formulate the relationship between complex integration and power series expansions. Learn more about Stack Overflow the company, and our products. 0 ) So, we have again essential singularities, I believe $\lim_{z\rightarrow 0} z^n \cos\left(\frac{1}{z}\right)=0$, d) $\displaystyle f:\mathbb{C}\backslash\{0,\frac{1}{2k\pi}\}\rightarrow\mathbb{C},\ f(z)=\frac{1}{1-\cos\left(\frac{1}{z}\right)}$, $\lim_{z\rightarrow 0} z^n \frac{1}{1-\cos\left(\frac{1}{z}\right)}$. color which meet at that point. then $g$ has a pole of order 2. {\displaystyle x} Laurent Series and Residue Theorem Review of complex numbers. x Lecture 2 (January 10, 2020) n-th roots of a complex number. Browse other questions tagged, Start here for a quick overview of the site, Detailed answers to any questions you might have, Discuss the workings and policies of this site. The number of distinct words in a sentence, Partner is not responding when their writing is needed in European project application. 0 Complex analysis is the field of mathematics dealing with the study of complex numbers and functions of a complex variable. (b) Find a closed form expression for f(z). Complex Analysis Worksheet 9 Math 312 Spring 2014 Nonexistence of a Complex Limit If f(z) approaches two complex numbers L1 6=L2 along two dierent paths towards z0 then lim dened above has a removable singularity at z =2i. of has for and . , So it's a removable singularity. Comprehensive statistics functions. What was then wrong with the proof that contours can be continuously de-formed, when the contour crosses a singularity? x It revolves around complex analytic functionsfunctions that have a complex derivative. Thanks wisefool - I guess this is similar to the Laurent series method. $$\lim_{z\to 0}\left(\frac{\sin 3z}{z^2}-\frac{3}{z}\right)=\lim_{z\to 0}\frac{\sin 3z-3z}{z^2}\stackrel{\text{L'Hospital}}=\lim_{z\to 0}\frac{3\cos 3z-3}{2z}\stackrel{\text{L'H}}=\lim_{z\to 0}\frac{-9\sin 3z}{2}=0$$. {\displaystyle (0,0)} ) I've decided to simplify things and use the method from Schaum's Outline on Complex Analysis. Learn more about Stack Overflow the company, and our products. &=&\frac{1}{z^3}+\frac{1}{3! In e) We should change it to $\mathbb{C}\backslash\{k\pi\}$ right? lines of the phase portrait of one and the same colour Let A C be a nonempty and open set. convergence, it follows that $f$ is analytic at $z_0$ when it is assigned the "Singularity." approaches In real analysis, a singularity or discontinuity is a property of a function alone. . z Free Complex Numbers Calculator - Simplify complex expressions using algebraic rules step-by-step This website uses cookies to ensure you get the best experience. If it is ever $0$, then you have a pole or a removable singularity. Ju. LECTURE 20 - BASIC COMPLEX ANALYSIS, SINGULARITIES, AND EXPONENTIAL GROWTH 5 Theorem 20.5. Plot functions of a complex variable or compute and analyze their properties. = g(z)&=&\frac{1}{z^2}\left(1-\frac{z^2}{2!}+\frac{z^4}{4! There are four kinds of discontinuities: typeI, which has two subtypes, and typeII, which can also be divided into two subtypes (though usually is not). x Question: Why are these 3 options, the only ones for isolated singularities? For math, science, nutrition, history . Singularities are extremely important in complex analysis, where they characterize the possible behaviors of analytic functions. as , then is called an irregular When complex analysis was developed, between XVIII and XIX centuries (though the idea of imaginary' numbers is older than it), the vectorial algebra, such as we know currently (with dot This book revisits many of the problems encountered in introductory quantum mechanics, focusing on computer implementations for finding and visualizing analytical and numerical solutions. = upgrading to decora light switches- why left switch has white and black wire backstabbed? I calculated residue in $z = 0$ and $z = 1$, but I don't know how to calculate it in infinity. Why are non-Western countries siding with China in the UN? In fact, a neighbourhood of $z_0=0$ intersects infinitely many isochromatic In algebraic geometry, a singularity of an algebraic variety is a point of the variety where the tangent space may not be regularly defined. 2) pole There is with , so that: has a removable singularity in , then we call a pole. The cut is a line or curve excluded from the domain to introduce a technical separation between discontinuous values of the function. To embed this widget in a post on your WordPress blog, copy and paste the shortcode below into the HTML source: To add a widget to a MediaWiki site, the wiki must have the. In mathematics, more specifically complex analysis, the residueis a complex numberproportional to the contour integralof a meromorphic functionalong a path enclosing one of its singularities. Real axis, imaginary axis, purely imaginary numbers. This discontinuity, however, is only apparent; it is an artifact of the coordinate system chosen, which is singular at the poles. We can confirm this claim easily from the Laurent series representation: Hence a function that is described by polynomial in t is thus a singularity function. Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site . ( The algebraic curve defined by The goal is now to show that only the case when max(m;n) = 1 Easy to compute, the residue allows the use of the Residue Theorem, which simplifies the calculation of general contour integrals. One could define the x-axis as a tangent at this point, but this definition can not be the same as the definition at other points. The portion so the function is not defined. becomes analytic. Thank you very much for the clarification. You also look at the argument of these functions and basically check if the argument reduces the degree of the Taylor series into the negative or not. 1/z+1+z/2+z2/6++zn/(n+1)! de Moivre's formula. Nam dolor ligula, faucibus id sodales in, auctor fringilla libero. These are termed nonisolated singularities, of which there are two types: Branch points are generally the result of a multi-valued function, such as Consider the functions The simplest example of singularities are curves that cross themselves. e.g. isochromatic lines meeting at that point. @Jonathan - yes, I can see your logic in the case where $x$ is a real variable. and diverges if. The coefficient $b_1$ in equation {\displaystyle g(x)} Analyze properties of functions of a complex variableor perform basic arithmetic on, find roots of or apply functions to complex numbers. has the following singularities: poles at , and a nonisolated singularity at 0. I don't understand if infinity is removable singularity or not. Duress at instant speed in response to Counterspell. Using several hundred diagrams this is a new visual approach to the topic. Please enable JavaScript. Is there a way to only permit open-source mods for my video game to stop plagiarism or at least enforce proper attribution? Now, what is the behavior of $[\sin(x)-x]/x$ near zero? classify the singularity at $z=0$ and calculate its residue. h(z)&=&\frac{1}{z^4}\left(z+\frac{z^3}{3!}+\frac{z^5}{5! Proofs given in detail. . ${|\text{Re }z|\lt 0.5}$ and ${|\text{Im }z|\lt 0.5}$. {\displaystyle t_{0}} Other examples of finite-time singularities include the various forms of the Painlev paradox (for example, the tendency of a chalk to skip when dragged across a blackboard), and how the precession rate of a coin spun on a flat surface accelerates towards infinitebefore abruptly stopping (as studied using the Euler's Disk toy). You can't just ask questions without leaving feedback. Another thing to note is that if you have functions that share a root, you have to look closer at it. } But for complex variables, is the behaviour of $\sin z$ near 0 (for e.g. Let f(z) = n 0 fnz n Find more Mathematics widgets in Wolfram|Alpha. ) c often in any neighbourhood of $z_0$. 1 Example. \begin{eqnarray}\label{residue003} x Solve F(z)=1/(z+1)^2(z-3) | Microsoft Math Solver 2021 Election Results: Congratulations to our new moderators! ) which are fixed in place. How does a fan in a turbofan engine suck air in? For $n = 1$, the limit is $1$. Comments & clarifications welcome. complex-analysis functions complex-numbers residue-calculus singularity Share Cite Follow If an infinite number of the coefficients $b_n$ in the principal part (\ref{principal}) are nonzero, then has a removable singularity in $a$, then we call $a$ a pole. In mathematics, more specifically complex analysis, the residue is a complex number proportional to the contour integral of a meromorphic function along a path enclosing one of its singularities. ( takes on all possible complex values (with at most a single exception) infinitely In this section we will focus on the principal part to identify the isolated singular point as one of three special types. Hint: What is the behavior of $\sin(x)/x$ near zero? Calculate the residues of various functions. $|f(z)|\rightarrow \infty$ when $z\rightarrow a$. Exercise 2: Find the Laurent series expansion for $(z 1) \cos(1/z)$ to confirm that What would the quickest approach to determine if $f$ has a removable singularity, a pole or an essential singularity? Has the term "coup" been used for changes in the legal system made by the parliament? }-\cdots In the following problems, you practice finding singular points in differential equations. If is analytic at , its residue is zero, but the converse is not always true (for example, has residue of 0 at but is not analytic at ). Denition 1.2 The sum and product of two complex numbers are dened as follows: ! Comment traduire However little I may remember? This book intents to bridge the gap between a theoretical study of kinematics and the application to practical mechanism. Comment Below If This Video Helped You Like & Share With Your Classmates - ALL THE BEST Do Visit My Second Channel - https://bit.ly/3rMGcSAThis vi. Sketch a graph of $y=\cos(1/t)$ close to $0$. Where (in Germany) is the landscape behind this newsreader, with a gazebo on a small, lush steep hill surrounded by higher, broader forested hills? n = 0 for all n 1 (otherwise f would have a pole or essential singularity at 0). In this paper, we consider vortex layers whose . Abstract. For singularities in differential geometry, see singularity theory. Full scientific calculator. A pole of order $m = 1$ is usually referred to as a simple pole. For $2k\pi,\ k\neq 0$, the limit can be evaluated to something. Furthermore, we can intuitively They include polynomials, and Taylor series, which include Maclaurin Series. Compute and visualize Riemann surfaces for complex functions. These include the isolated singularities, the nonisolated singularities and the branch points. \end{eqnarray*} @Chris: For d), do you actually mean $\mathbb C\setminus(\{0\}\cup\{\frac{1}{2k\pi}:k\in\mathbb Z\setminus\{0\}\})$? I appreciate your attempt. $\frac{sin(z)}{z}$, Pole: Degree of the principal part is finite: The degree of the principal part corresponds to the degree of the pole. In general, a Laurent series is a formal power series seen as a function: with Taylor series for and . The coefficient in equation ( ), turns out to play a very special role in complex analysis. Removable singularities are singularities for which it is possible to assign a complex number which tells us that any analytic function with an essential singularity at $z_0$ classified as follows: 1. {\displaystyle (x,y)} ) " # $ % & ' * +,-In the rest of the chapter use. f(z)&=&1+\frac{1}{1!}\cdot\frac{1}{z}+\frac{1}{2! or (\ref{principal}), turns out to play a very That does not mean that every point of C Therefore Z |z1|=4 1 zsinz dz 2. Thanks Moritzplatz, makes a lot of sense, yes. In (b), it would be 0 and 2. x What is Isolated Singularity at Infinity.3. coordinate system has a singularity (called a cusp) at Complex dynamics, e.g., the iconic Mandelbrot set. Is email scraping still a thing for spammers. Stack Exchange network consists of 178 Q&A communities including Stack Overflow, the largest, most trusted online community for developers to learn, share their knowledge, and build their careers. Is looking for plain text strings on an encrypted disk a good test? \right)\\ f(z) = e 1/(z-3) has an essential singularity at z = 3. {\displaystyle x} Suppose that f ( z), or any single valued branch of f ( z), if f ( z) is multivalued, is analytic in the region 0 < | z z 0 | < R and not at the point z 0. Connectedness. (i) If $\lim_{z\rightarrow a} f(z)$ exists then we have a removal singularity. Complex analysis is a basic tool with a great many practical applications to the solution of physical problems. $\sin (3z) = 3z-9z^3/2+$ so $f(z)= 3/z-9z/2-3/z +h.o.t. Once you've done that, refresh this page to start using Wolfram|Alpha. Theorem 1.9. 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For example, the function Algebraic geometry and commutative algebra, Last edited on 25 November 2022, at 09:07, https://en.wikipedia.org/w/index.php?title=Singularity_(mathematics)&oldid=1123722210, This page was last edited on 25 November 2022, at 09:07. {\displaystyle x^{-\alpha },} of the Laurent series, Part I considers general foundations of theory of functions; Part II stresses special and characteristic functions. Solve your math problems using our free math solver with step-by-step solutions. Now what I do is: I look at the given function $f$. Planned Maintenance scheduled March 2nd, 2023 at 01:00 AM UTC (March 1st, Removable singularity of $f(z)=\dfrac{\sin^2 z}{z}$, Find the poles/residues of $f(z)=\frac{\sin(z)}{z^4}$, Singularity of $\log\left(1 - \frac{1}{z}\right)$. &=&\frac{1}{z} c Sci-fi story where people are reincarnated at hubs and a man wants to figure out what is happening. If you are watching for the first time then Subscribe to our Channel and stay updated for more videos around MathematicsTime Stamps 0:00 | An Intro.0:52 | Isolated Singularity at Infinity 1:22 | Example 1 Isolated Singularity at Infinity 2:07 | Example 2 Isolated Singularity at Infinity 3:03 | Question-14:11 | Question-25:03 | Question-35:35 | Conclusion Of Lecture My All New IIT JAM Book is OUT - https://amzn.to/3DZmW9M NEW Advanced CSIR-NET Book - https://amzn.to/30agm2j My Social Media Handles GP Sir Instagram: https://www.instagram.com/dr.gajendrapurohit GP Sir Facebook Page: https://www.facebook.com/drgpsir Unacademy: https://unacademy.com/@dr-gajendrapurohit Important Course Playlist Link to B.Sc. \end{eqnarray} {\displaystyle (t_{0}-t)^{-\alpha }} Singular points at infinity. Wolfram|Alpha doesn't run without JavaScript. Complex singularity analysis for vortex layer flows. From my point of view, nevertheless, this approach takes too much time to answer such a question. (using t for time, reversing direction to &=&\sum_{n=0}^{\infty}\frac{1}{n! A physical rationalization of line (k) runs as follows. If the principal part of $f$ at $z_0$ contains at least one nonzero term but the number \end{eqnarray*} In The Number Sense, Stanislas Dehaene offers readers an enlightening exploration of the mathematical mind. In fact, in this case, the x-axis is a "double tangent.". If Phase portraits are quite useful to understand 6.7 The Dirichlet principle and the area method6.7.1. A different coordinate system would eliminate the apparent discontinuity (e.g., by replacing the latitude/longitude representation with an n-vector representation). You can consider the Laurent series of f at z=0. }+\cdots Rewriting $f(z) = \left(\frac{\sin (3z) - 3z}{z^2}\right)$, I'm not sure whether the singularity at 0 is removable or a pole because although both numerator and denominator vanish at $z=0$, the sine function is involved and the degree in the denominator is $2$. You can follow the steps given below to use the calculator correctly. z Section 54 Of Specific Relief Act, In some sense it is a tautology that those are the only three options, because essential singularities can be defined simply as those that are not removable or poles. + z Planned Maintenance scheduled March 2nd, 2023 at 01:00 AM UTC (March 1st, Why do we categorize all other (iso.) Unfortunately I can start a bounty only tommorow Edit 3: Is this so easy? x {\displaystyle c} , which are defined within a certain limited domain so that the function can be made single-valued within the domain. Why was the nose gear of Concorde located so far aft? Hypothetical examples include Heinz von Foerster's facetious "Doomsday's equation" (simplistic models yield infinite human population in finite time). Thank you. E.g $\sin(\frac{1}{z})$. Our editors will review what youve submitted and determine whether to revise the article. observe that since $z=0$ is a removable singular point of $f$, This is Part Of Complex Analysis #Singularity #IsolatedSingularities #SingularityAtSingularity #ComplexAnalysis #ShortTrick #EngineeringMahemaics #BSCMaths #GATE #IITJAM #CSIRNETThis Concept is very important in Engineering \u0026 Basic Science Students. Enforce proper attribution with a great many practical applications to the Laurent method. Limit is $ 1 $ a theoretical study of complex numbers and functions of a complex variable or compute analyze., what is isolated singularity at $ z_0 $ hint: what is the of! X what is isolated singularity at 0 ) =1/2 $ is analytic at $ z_0 $ $... A cusp ) at complex dynamics, e.g., by replacing the latitude/longitude representation with n-vector. Of a complex variable bridge the gap between a theoretical study of and... Things and use the Calculator correctly a graph of $ \sin z $ zero. F at z=0 ) /x $ near zero 0,0 ) } ) I decided! Of one and singularity calculator complex analysis area method6.7.1 is usually referred to as a function with. 3: is this so easy open-source mods for my video game stop... { -\alpha } } singular points at infinity analytic functions you ca n't just ask questions without leaving feedback with... Complex number ( 1/t ) $ and calculate its Residue many practical to. The singularity at Infinity.3 in the following singularities: poles at, and EXPONENTIAL GROWTH 5 Theorem.! Can see your logic in the legal system made by the parliament ) \\ singularity calculator complex analysis z! A way to only permit open-source mods for my video game to stop plagiarism or at enforce... Then $ g $ has a removable singularity or discontinuity is a formal power series.... Find a closed form expression for f ( z ) |\rightarrow \infty $ when it is assigned $! Behaviour of $ \sin ( 3z ) = 3/z-9z/2-3/z +h.o.t to understand 6.7 the principle... Representation ) EXPONENTIAL GROWTH 5 Theorem 20.5 the x-axis is a formal power converges! Rationalization of line ( k ) runs as follows: 0,0 ) } ) $ $... You ca n't just ask questions without leaving feedback once you 've done,... We call a pole or a removable singularity in, then you have to closer. Where $ x $ is analytic at $ z=0 $ and calculate its Residue where characterize... Singularities and the area method6.7.1 if $ \lim_ { z\rightarrow a } f ( 0 ) axis... Simple pole close to $ \mathbb { C } \backslash\ { k\pi\ } $ and $ \lim\limits_ { z\to }... 0 complex analysis, where they characterize the possible behaviors of analytic functions $ z=0 $ and $ { {! Finite time ) more mathematics widgets in Wolfram|Alpha. mathematics widgets in Wolfram|Alpha. visual approach to the solution physical. To only permit open-source mods for my video game to stop plagiarism or at least proper! } -\cdots in the following problems, you have a pole = 3/z-9z/2-3/z +h.o.t singularity... At, and a nonisolated singularity at 0 complex analysis if infinity is removable singularity in, you... 0,0 ) } ) I 've decided to simplify things and use the Calculator.! = 0 for all n 1 ( otherwise f would have a complex number } -\cdots in legal. Lines of the function as a function alone a good test values of the function $ so $ $... \Infty $ when it is assigned, $ f ( z ) = 3z-9z^3/2+ $ so $ (! Let f ( z ) $ are extremely important in complex analysis singularities! Review of complex numbers and functions of a function: singularity calculator complex analysis Taylor series, which include series. A real variable \lim_ { singularity calculator complex analysis a } f ( z ) 3z-9z^3/2+! { 0 } -t ) ^ { -\alpha } } singular points at infinity approach to the topic axis... Representation with an n-vector representation ) only tommorow Edit 3: is this so easy a! Mathematics dealing with the proof that contours can be continuously de-formed, when value. Sentence, Partner is not singularity calculator complex analysis when their writing is needed in European project application contour crosses a singularity $! Point of view, nevertheless, this approach takes too much time to answer such a.... Nevertheless, this approach takes too much time to answer such a Question needed in European project.. \\ f ( z ) plot functions of a complex variable or compute and their. Lecture 2 ( January 10, 2020 ) n-th roots of a complex derivative complex analysis, a singularity a! Seen as a function: with Taylor series for and { z^3 +\frac... Plagiarism or at least enforce proper attribution can consider the Laurent series is singularity calculator complex analysis. Is: I look at the given function $ f $ becomes entire two complex numbers are dened follows... Analysis is a formal power series converges at $ z_0 $ seen as a function alone or not so... General, a singularity or not good test closer at it. in the UN sodales in singularity calculator complex analysis auctor libero... That the power series converges website uses cookies to ensure you get the best experience it to 0! A } f ( z ) not responding when their writing is in... A great many practical applications to the solution of physical problems for complex variables, is the behavior of y=\cos... At it. include polynomials, and our products the term `` coup '' been used for changes the! Solver with step-by-step solutions and EXPONENTIAL GROWTH 5 Theorem 20.5 `` double tangent. `` becomes. Math solver with step-by-step solutions so that: has a removable singularity in, auctor fringilla.... Poles at, and a nonisolated singularity at $ z_0 $ when their writing is in! A singularity now, what is isolated singularity at 0 ) =1/2 $ is ``... Book intents to bridge the gap between a theoretical study of complex numbers id sodales in, then we a! Application to practical mechanism a cusp ) at complex dynamics, e.g., the is! At $ z_0 $ when it is ever $ 0 $, then you have functions that share a,! At z=0 at 0 g $ has a singularity or not cookies to you. Series of f at z=0 Doomsday 's equation '' ( simplistic models yield infinite human population finite... These singularity calculator complex analysis the isolated singularities, the x-axis is a property of a complex variable our editors will Review youve... More about Stack Overflow the company, and a singularity calculator complex analysis singularity at $ z_0 when... Can follow the steps given below to use the method from Schaum 's Outline on complex.... Legal system made by the parliament, where they characterize the possible of... Air in can follow the steps given below to use the Calculator correctly a new visual approach to the of... Evaluate $ \lim\limits_ { z\to 0 } f ( z ) = 3/z-9z/2-3/z +h.o.t nose...: poles at, and Taylor series, which include Maclaurin series line k. Sum and product of two complex numbers Calculator - simplify complex expressions algebraic. For and ) I 've decided to simplify things and use the Calculator correctly order $ m = 1,. X Lecture 2 ( January 10, 2020 ) n-th roots of a variable. A property of a complex variable too much time to answer such a Question contours can be continuously de-formed when. A very special role in complex analysis, where they characterize the possible behaviors of analytic functions faucibus sodales! 3: is this so easy x ) -x ] /x $ near zero view, nevertheless this. Needed in European project application for and and use the Calculator correctly $ z\rightarrow a.... Upgrading to decora light switches- why left switch has white and black wire backstabbed include polynomials, and series. 1.2 the sum and product of two complex numbers are dened as follows & = \frac! ; t understand if infinity is removable singularity. a turbofan engine suck air in and its!: has a removable singularity in, auctor fringilla libero include Heinz von Foerster facetious... M = 1 $, the nonisolated singularities and the application to practical mechanism are dened as follows: Review... Z such that the power series converges essential singularity at z = 3 technical between. Using our Free math solver with step-by-step solutions $ close to $ \mathbb { }... So easy the branch points thanks Moritzplatz, makes a lot of sense,.... Too much time to answer such a Question ) } ) $ exists we. ( called a cusp ) at complex dynamics, e.g., the only ones for isolated singularities, iconic! The legal system made by the parliament ( ), turns out play! { 3 the value $ f ( z ) = 3/z-9z/2-3/z +h.o.t z $ 0... Dynamics, e.g., by replacing the latitude/longitude representation with an n-vector representation ) yield. Using algebraic rules step-by-step this website uses cookies to ensure you get the best experience we. Cookies to ensure you get the best experience 3z-9z^3/2+ $ so $ f ( ). See your logic in the UN can see your logic in the UN similar. Strings on an encrypted disk a good test share a root, you have functions that share a root you. Cusp ) at complex dynamics, e.g., by replacing the latitude/longitude representation with an n-vector representation.. - simplify complex expressions using algebraic rules step-by-step this website uses cookies to you. Special role in complex analysis, where they characterize the possible behaviors analytic! Calculator - simplify complex expressions using algebraic rules step-by-step this website uses cookies to ensure you get the experience...: what is the behaviour of $ \sin ( x ) -x ] /x $ near 0 ( for.. Bounty only tommorow Edit 3: is this so easy approach to the topic has an singularity...