• Omid m.kandelusy added an answer:
    Does any one know how to compute definite integrals involving zero oreder first kind modified bessel function ?

    i want to compute integral of f(x) which is defined as

    f(x)=e(-ax)* I_0(b*(x^ 0.5))


    I_0 is zero order first kind modified bessel function.

    and, the integral is done from x=0 to x=+\infinity

    i know the answer in the case of zero order first kind bessel finction (J_0), but im not sure to use the transformation from unmodified to modified. i would be really grateful to have your comments.

    Omid m.kandelusy · Babol Noshirvani University of Technology

    thanks Dear Lichtblau

    yes, Mathematica is great

    in fact, i just work with Matlab and Mathematica, but in my research, i need to analytically derive some expressions myself.

  • Andrew Messing added an answer:
    Why should we teach improper integrals?

    I've taught using maybe a dozen calculus textbooks (not through my own choice) and examined many more, and the only ones that do not include improper integrals are those that I've never heard of anybody using. Yet

    1) Improper integrals do not generalize. That is, once we leave the real number line and start working in Rn, improper integrals do not work (indeed, as defined they make no sense).

    2) The only pedagogical function they might serve has to do with convergence, but textbooks invariably include the so-called "integral test" anyway and anything of value for understanding convergence/divergence we obtain from first learning of improper integrals could be introduced merely with the "integral test".

    3) By the time most students get to improper integrals, integration has come to be understood (at least almost) entirely in terms of finding antiderivatives and using the "fundamental theorem" of calculus. Thus improper integrals represent a conceptual break from previous work with integration. Also, as whatever failings the Riemann integral may have in terms of what functions it can't deal with, Lebesgue integrals (and others) will do all that improper integrals can and more.

    4) Improper integrals are deceptively (even for advanced students) tricky if not absolutely convergent.

    So what's the justification for using them to teach elementary calculus?

    Andrew Messing · Harvard University

    Dear Erkki:

    Thanks for the link!  Not just for its contents but for the context you gave. In any given field at any given time, it seems like there are a few places where, if one has particular interests, that's the place to be. Unfortunately, there isn't exactly a directory containing a list "cool institutes producing research  you'll really want to keep an eye on". One finds such places through networking or reading a good deal of research and realizing that much of the neatest stuff is all coming from e.g., the Santa Fe Institute, or the Isaac Newton Institute for Mathematical Sciences. Thanks to you I can add Uppsala University's "Department of Physical and Analytical Chemistry" to the list of departments, labs, centers, etc., producing research that I should follow because of their work and my interests. Much appreciated!


  • Praveen Agarwal added an answer:
    Is the order of coordinates important in the calculus of tensors?

    For example, if I use spherical coordinates, r is X1, theta is X2 and phi is X3. But could I use, for example, phi as X1, theta as X2 and r as X3 in the calculus of tensors?

    Praveen Agarwal · Anand International College of Engineering

    I also agree with Prof. Hady

  • Arturo Tozzi added an answer:
    Any suggestion to represent "thinking" in mathematical language or logic?

    How could we describe the act of "thinking" with mathematical tools? Which paradigm is best suited for? What does "thought" mathematically mean? Is there any alternative to the procedural (linear) conception of neural calculus? 

    Arturo Tozzi · Azienda Sanitaria Locale Napoli 2 Nord

    See also: 


  • Olaniyi Samuel Iyiola added an answer:
    Riemann-Liouville fractional integral operator I_(a+ ) is defind for x>a but what happens if x=a?

    Riemann-Liouville fractional integral operator Ia+(x)  is defined for x>a but what happens if x=a?

    Olaniyi Samuel Iyiola · King Fahd University of Petroleum and Minerals, KFUPM

    The question is interesting. I have also been working on fractional derivatives for some time now. You can find answers to some of thess issues if you have access to this book: 

    A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo. Theory and Applications of
    Fractional Differential Equations, volume 204 of Mathematics Studies. Elsevier,
    Amsterdam, 2006.

    I use it alot together with 

    I. Podlubny. Fractional Differential Equations, volume 198 of Mathematics in
    Science and Engineering. Acad. Press, 1999.

    They contain some of the conditions under which thess kind of limits exist.

    Hope it would be useful. By the way, good responses from Mohamed and Jukka.

  • James F Peters added an answer:
    Should we teach limit or derivative first?
    Should we teach limit as a tool for derivation or should we teach in depth as a subject before derivative concept?
    James F Peters · University of Manitoba

    Dear Costas,

    Your textbook is a treasure!   Many thanks for sharing it.    Do you still use this book in your teaching?    The Escher drawings fit perfectly with your approach.    I wish I had started by taking your class.


  • Anna Valkova Tomova added an answer:
    Can anyone tell me how to check the convergence and divergence of the function like integral(1/{Sqrt(x^4-1)}), with lower limit 2 and upper limit infinity?
    I also need to check for which real of p the integrals converge or diverge
    Integral(|x|^p), with lower limit -1 and upper limit 1.

    My answer for the first definite integral and for the second indefinite integral are done by http://www.wolframalpha.com/ on the following figures. Your sincerely: Anna Tomova.

  • Alexandre Costa added an answer:
    Is a histopathologic or cytologic confirmation of malignancy really needed for suspicious biliary stricture before surgery?
    A biliary stricture (common bile duct or confluence) without history of calculi is often proposed for exploratory surgery due to low sensivity of bioptic procedures and the need to avoid further bilirubin rise.
    Alexandre Costa · Centro Hospitalar de Vila Nova de Gaia/Esphino

    Yes, pre-operative cytologic or, preferably, histologic confirmation is the ideal, and new biopsy devices may help - but still are a little too far from being reliable. Unfortunately, not always we can obtain that information and, if those procedures results are negative for malignancy, we have to rely on the clinical presentation and, to some extent, abdominal MRI-MRI cholangiopancreatography. Nevertheless, when in doubt, I think it is in the best interest of the patient to propose a surgical approach so an opportunity for a cure not be missed (albeit all the negativity of cytology or biopsy). 

  • DO you have any scalar- and vector field modeling software recomendations?

    I would like to know what standard softwares are normally used to model and visualize scalar and vector fields on a PC.

    Efraín Antonio Domínguez Calle · Pontifical Xavierian University (Bogota)

    I love to recommend python programming language for this kind of tasks.Python has specialized modules for modelling and visualization (from simple to complex). Python is totally free and open source, I suggest Anaconda python to be download instead of installing python interpreter and then download and plug modules, this will save your time. As IDE I recommend pycharm community edition but the alternatives are huge. For very advanced 3d plots Mayavi module is the best but you also can use matplotlib and Chaco (see the links).

  • Richard Herrmann added an answer:
    Is there any geometrical relation between fractional calculus and fractal ?
    Of course there are some concepts of fractional calculus that are co-related to fractal. But how are they geometrically related ?
  • Sanjeev Gupta added an answer:
    Is there any set-valued map which is not satisfy any properties of subdifferential calculus?

    I'm looking for optimizing multivalued  vector valued function.

    Sanjeev Gupta · Indian Institute of Technology Kanpur

    Dear Sir my problem is associated with the nonsmooth convex vector optimization problem. I want to give an concrete example of set-valued which is not the subdifferential map.

    Thanks for your kind anticipation. But my problem is not solve yet now. Kindly give some suggestion on this question.

  • Hammouch Zakia added an answer:
    Is there a definition of the Mittag-leffler function for variable-order fractional calculus?
    E_q(t) with q a bounded function.
    Hammouch Zakia · FSTE Université Moulay Ismail

    Dear Professor  R.Heremann,

    Thank you !

  • Richard Herrmann added an answer:
    What does it mean to the fractional calculus if the constant is equal to zero in some definitions and not equal to zero in another's definition?
    Riemann–Liouville fractional derivative & Caputo fractional derivative & Modified Riemann–Liouville
  • Richard Herrmann added an answer:
    Is it possible to apply the Leibniz product rule for variable-order fractional derivation?
    D_q(t) (f.g) = D_q(t) f .g + f.D_q(t)g
  • Anna Valkova Tomova added an answer:
    What is integral of (secx)^3 wrt x ?
    If limits considered from 0 to (pi/4) and without limits also.

    My answer from wolfralpha.com is in the file attachment. Your sincerely; Anna Tomova

  • Anatolij K. Prykarpatski added an answer:
    What do you think about the definition of *-differentiation which is given via "Generalized Runge-Kutta method with respect to non-Newtonian calculus?

    The non-Newtonian calculi were created in the period from 1967 to 1970 by Michael Grossman and Robert Katz. In July of 1967, they created an infinite family of calculi that includes the classical calculus, the geometric calculus, the harmonic calculus, and the quadratic calculus. In August of 1970, they created infinitely-many other calculi, including the bigeometric calculus, the biharmonic calculus, and the biquadratic calculus. All of the calculi can be described simultaneously within the framework of a general theory. They decided to use the adjective "non-Newtonian" to indicate any of the calculi other than the classical calculus.

    M. Grossman sent a letter to R. Katz on 21 July 2014; he said that

    "As you well know, for many years lots of people, especially various pure mathematicians, claimed that our work was useless. But, despite their discouraging and sometimes arrogant comments, we always knew that NNC has considerable potential for application in science, engineering, and mathematics. And we were right!! ".

    I agree strongly with you Professor.

    Anatolij K. Prykarpatski · AGH University of Science and Technology in Kraków

    Dear Ugur, thanks - yet I guess you are slightly mislead concerning my  in no sense critical comments (being strongly math backgrounds educational, in particular concerning   things, which are related to general of analytical nature   math problems - as you well, I am sure, understand too - that any  such a math problem is not solvable on the whole, if it does not allow some  either local  or global linearization. - Even more, the math analysis as science would not exist,  if there were no possibility to linearize a mapping in some  vicinity of a domain point! )  - As you can also easily observe,  my  clearly formulated  intentions  were completely aimed at  the pure mathematical essence of these "calculi" activity subject to this fundamental linearity property,  and concerned in no sense their , eventually,  possible applications! Why not? - -nobody knows!

    All the best and regards!

  • Michael Livshits added an answer:
    What is your opinion that students find difficult in learning introductory calculus?
    Traditional teaching has shadowed the current learning of calculus.
    Michael Livshits · mathfoolery

    @Andrew Messing, it looks easy until you try, and when you try you will understand how difficult it is, especially in our real world, not in the ideal world that you may be imagining while discussing this or that "improved" approach. And if you tell me how "dumping the Riemann integral" or "rigorous formulation of infinitesimals" would make introductory calculus more understandable, I will be all ears.

  • George Stoica added an answer:
    What is difference between greatest value and maximum value of a real valued function? When do they coincide?
    Real valued functions
    George Stoica · University of New Brunswick

    Dear Suruchi,

    Max is the largest number within a set, whereas sup bounds the set from above. Sup may or may not be part of the set itself. If sup is part of the set, it is also the max. Examples are straightforward.



  • Roman Sznajder added an answer:
    When is the power series of an even function alternating?

    This is motivated by a calculation using Fourier transforms of filters / mollifiers. Since it is such a classical question, I suspect there is a classical answer out there and would appreciate help satisfying my curiosity.

    Roman Sznajder · Bowie State University

    Dear Bill:

    If I understand it correctly, you are expanding a function f in Taylor series at a=0, which is natural, as the function is even. As such, all odd-order derivatives of f at 0 must be zero. Thus, we deal only with f^(2k)(0) and this sequence should alternate. That's it. It seems not to exist any reasonable classification of functions with this property. 

    All the best,


  • Abedallah M Rababah added an answer:
    Who first defined orthogonal polynomials?
    It is known that Legendre and Chebyshev have contributions in the field.
    Abedallah M Rababah · Jordan University of Science and Technology

    Dear @Professor Afag Ahmad, your contribution is highly appreciated. I want to know in particular who did first define orthogonal polynomials.

  • R. C. Mittal added an answer:
    What do you think about the non-Newtonian calculi which have considerable potential for use as alternatives to the classical calculus?

    "The non-Newtonian calculus is a self-contained system independent of any other system of calculus. Therefore anyone may be surprised to learn that there is a uniform relationship between the corresponding operators of the non-Newtonian calculus and the classical calculus" (Grossman and Katz_1978)

    R. C. Mittal · Indian Institute of Technology Roorkee

    Dear Dr. Kadak, thanks for giving me this link. It is a good paper and some new ideas for me.

  • Rogier Brussee added an answer:
    Can a multivariate function f(x,y,..) be converted to a lower dimension function?

    I am trying to find if there is a mathematical formula that can, for example, convert a  function f(x,y) to f(r) such that f(x0,y0) = f(r0) and f(x+a,y+a) = f(r+a).


    Rogier Brussee · Hogeschool van Utrecht

    Well you can do

    Map(X_1 \times X_2 \times X_3 , Y)

               \iso Map(X_1, Map(X_2\times X_3, Y))

                \iso Map(X_1, Map( X_2, Map(X_3, Y)))

    I think in some functional programming languages (Haskell IIRC) they even do this in practice, so  they write the "Curried" form

    f: int -> string -> bool -> double

    which means f: int -> (string -> (bool -> double)))

    for what in most computer languages you would write as 

    f(int, string, bool) -> double 

    or C/Java  style

    double f(int, string, bool)

  • James F Peters added an answer:
    How can we construct surfaces of revolution in Cartan-Vranceanu Space?

    How can we construct surfaces of revolution in Cartan-Vranceanu Space? Frankly, we should define a rotation matrix, and its determinat must be zero. But, I have difficulties to find that matrix. Thank you.

    James F Peters · University of Manitoba

    This is a good question.   Earlier I gave an answer to this question but for some reason RG did not record my answer. 

    A start on answering this question can be found in

    A. Balmus, Differential Geometry--Dynamical Systems, 2009:


    There are 20 places in this monograph where surfaces are revolution, starting with Example 3.17 (Biharmonic curves on a surface of revolution), p. 49.   See Example 3.18 (Biharmonic curves of Cartan-Vranceanu spaces), starting on page 50.

  • Peter T Breuer added an answer:
    How can one show that the left limit is smaller than the right limit?

    In the figure lines are on the real axis.

    What I am interested in to prove that the left limit of the red interval is to the left of the left blue one and correspondingly for the right end.

    I was thinking that we can prove it by the definition of the left limit and right limit but I stuck with in it.

    Any suggestions to prove it mathematically will be highly appreciable.

    Peter T Breuer · Birmingham City University

    Somehow, I don't think we need to! (see Apostol).

    f(x) <= supx f(x)

    is good enough for me!

  • Ezequiel Soule added an answer:
    Can anyone suggest a method to solve the following constrained functional minimization problem?

    When a functional of one function is linear in the derivative of the function, I=int(f(y) + A(y)y'), the Euler-Lagrange equation leads to an algebraic equation as the terms involving y' cancel each other. But if the functional depends on more than one function this doesn´t happen:
    I=int(f(y,z) + A(y,z)y' + B(y,z)z'), the E-L equations are:

    df/dy + dB/dy*z' - dA/dz*z'=0
    df/dz + dA/dz*y' - dB/dy*y'=0

    These are first-order differential equations. Can this problem be solved, considering that there are at least two boundary conditions (imposed or natural)?

    Actually my problem comes from a functional I=int(f(y,z)), subject to a constraint linear in the derivatives, so the E-L equation of the augmented functional using the linear constraint lead to first order differential equation. BUT, if I re-write the linear constraint:

    A(y,z)y' + B(y,z)z' + C=0


    C/(A(y,z)y' + B(y,z)z') + 1=0

    and I construct the augmented functional with this constraint, the augmented functional is no longer linear in the derivatives and the E-L equation is a second order differential equation.
    Does it make sense? Can the same problem lead to differential equations of different orders? What happens then with the boundary conditions?

    Ezequiel Soule · Universidad Nacional de Mar del Plata

    Ha, it was much easier (and more obvious) than I thought... Thank you very much Vladimir!

  • Luiz C. L. Botelho added an answer:
    What is the physical significance of poles in Laurent's series?
    In Laurent's series expansion of complex functions we encounter a term pole.
    Luiz C. L. Botelho · Universidade Federal Fluminense

    Remember that a Laurent series is a Taylor series on the inverse variable .That is whole useful when understand one complex variable on the Rieman sphere , where analytycal functions around the infinity (essential singularities around the origem -or any other point by translation )must be analytic complex analysis understood 

  • Luiz C. L. Botelho added an answer:
    What are the applications of Ostrogradsky equation?

    While teaching calculus of variations, I came across the Ostrogradsky equation for extremizing functionals involving functions with many independent variables. What are the other applications of Ostrogradsky equation? Thanks. - Sundar

    Luiz C. L. Botelho · Universidade Federal Fluminense

    There is an important fourth order lagrangian in string theory :that relative to theextrinsic string .I think that imposing APROPRIATE AND CORRECT  boundary conditions on higher order elliptic problems is somewhat non trivial issue (Path integrals for higher order Lagrangians may be an interesting issue on the subject!)see

    1-J. T. Oden and J. N. Reddy, An Introduction to the Mathematical Theory of Finite Elements, John Wiley & Sons, New York, NY, USA, 1976. View at Zentralblatt MATH

    :2-ISRN High Energy Physics
    Volume 2012 (2012), Article ID 674985, 25 pages
    Research Article
    Basics Polyakov’s Quantum Surface Theory on the Formalism of Functional Integrals and Applications
    Luiz C. L. Botelho

  • Luisiana Cundin added an answer:
    How do you show that the integral is convergent or divergent?

    I have an integral and wanted to see whether it converges or diverges. I know there are many methods that can be used to show it.

    Note that -l(ell) to l(ell) belong to real numbers.

    I have the comparison method in my mind, but I am stuck with it. Any other suggestions will be highly appreciated. 

    Luisiana Cundin · Die Wand : leben heißt kampf

    Peter is right... But, complex analysis is becoming somewhat of a dinosaur these days of computer assisted knowledge. Anyway, a very good book for complex analysis and analytic expansion is Zeev Nehari's "Conformal Mapping", but it is out of print, although there are other books to get on the subject.

    The reason I suggested Widder's book is that he provides general theorems that either prove an integral either convergent or not, based on very loose and wide constraints. 

  • James F Peters added an answer:
    How can situation calculus and event calculus be integrated?

    Situation calculus and event calculus.

    James F Peters · University of Manitoba

    In addition to the incisive observations by @Abedallah M Rababah, the relationship between a discrete event calculus and the situation calculus is investigated in

    E.T. Mueller, Discrete event calculus with branching time, 2007:


    See Section 5, starting on page 6, where the equivalence of a branching discrete event calculus (BDEC) and the situation calculus is covered.   Perhaps you will find Section 7 (Actual situations and events), starting page 17, interesting.    In that section,  the BDEC is extended and used to distinguish between hypothetical and actual situations.    Also, Section 8 (Related work), starting on page 18, is very detailed.

About Calculus

The theory and application of differentiation, integration and limits.

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