Function Definition and 1000 Threads

In mathematics, a function is a binary relation between two sets that associates to each element of the first set exactly one element of the second set. Typical examples are functions from integers to integers, or from the real numbers to real numbers.
Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time. Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly enlarged the domains of application of the concept.
A function is a process or a relation that associates each element x of a set X, the domain of the function, to a single element y of another set Y (possibly the same set), the codomain of the function. It is customarily denoted by letters such as f, g and h.If the function is called f, this relation is denoted by y = f (x) (which reads "f of x"), where the element x is the argument or input of the function, and y is the value of the function, the output, or the image of x by f. The symbol that is used for representing the input is the variable of the function (e.g., f is a function of the variable x).A function is uniquely represented by the set of all pairs (x, f (x)), called the graph of the function. When the domain and the codomain are sets of real numbers, each such pair may be thought of as the Cartesian coordinates of a point in the plane. The set of these points is called the graph of the function; it is a popular means of illustrating the function.
Functions are widely used in science, and in most fields of mathematics. It has been said that functions are "the central objects of investigation" in most fields of mathematics.

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  1. SaschaSIGI

    I Understanding Hessian for multidimensional function

    Hello everybody, I have a question regarding this visualization of a multidimensional function. Given f(u, v) = e^{−cu} sin(u) sin(v). Im confused why the maximas/minimas have half positive Trace and half negative Trace. I thought because its maxima it only has to be negative. 3D vis 2D...
  2. chwala

    Find the derivative of the given function

    Let's see how messy it gets... ##\dfrac{dy}{dx}=\dfrac{(1-10x)(\sqrt{x^2+2})5x^4 -(x^5)(-10)(\sqrt{x^2+2})-x^5(1-10x)\frac{1}{2}(x^2+2)^{-\frac{1}{2}}2x}{[(1-10x)(\sqrt{x^2+2})]^2}##...
  3. I

    Show that the given function is decreasing

    As a follow up for : https://www.physicsforums.com/threads/let-k-n-show-that-there-is-i-n-s-t-1-1-k-i-1-2-k-i-1-4.1054669/ show that ## \alpha\left(k\right)\ :=\ \left(1-\tfrac{1}{k}\right)^{\ln\left(2\right)k}-\left(1-\tfrac{2}{k}\right)^{\ln\left(2\right)k} ## is decreasing for ##...
  4. Silvia2023

    For this Partial Derivative -- Why are different results obtained?

    Given a function F(x,y)=A*x*x*y, calculate dF(x,y)/d(1/x), to calculate this derivative I make a change of variable, let u=1/x, then the function becomes F(u,y)=A*(1/u*u)*y, calculating the derivative with respect to u, we have dF/du=-2*A*y*(1/(u*u *u)) replacing we have dF/d(1/x)=-2*A*x*x*x*y...
  5. PeaceMartian

    I What are the Zeta Function and the Riemann Hypothesis?

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  6. phos19

    I Fermi energy for a Fermion gas with a multiplicity function ##g_n##

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  7. E

    Vector is a function of its position or not?

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  8. MatinSAR

    Distance as a function of time for two falling stones

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  9. E

    Details regarding the high temperature limit of the partition function

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  10. Mohmmad Maaitah

    How to find intervals where this function is decreasing and increasing?

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  11. M

    Finding where this function is increasing or decreasing

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  12. M

    Python Graphing a piecewise function (Python)

    I am trying to write a python script to plot the function, Where ##V_0 = 5~V## ##t_0 = 10~ms## ##\tau = 5~ms## My script that I have written to try to do this is, Which plots, However, the plot is meant to look like this with the horizontal line. Can someone please give me some guidance to...
  13. L

    Discharging a capacitor -- Calculate the current as a function of time

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  14. F

    I Partial derivatives of the function f(x,y)

    Hello, Given a function like ##z= 3x^2 +2y##, the partial derivative of z w.r.t. x is equal to: $$\frac {\partial z}{\partial x} = 6x$$ Let's consider the point ##(3,2)##. If we sat on top of the point ##(3,2)## and looked straight in the positive x-direction, the slope The slope would be...
  15. T

    I Infinite product representation of Bessel's function of the 2nd kind

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  16. barryj

    I Formula for credit card balance as a function of payments

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  17. A

    Oven controller block diagram, transfer function and temperature calcs

    FIGURE 5 shows an electrically heated oven and its associated control circuitry. The current, I, to the oven's heating element is fed from a voltage-controlled power amplifier such that I = EK1. A voltage, VD, derived from a potentiometer, sets the desired oven temperature, TD. The oven...
  18. kakaho345

    Finding free electron gas Green function in Fourier space

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  19. J

    I Solution of delayed forcing function

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  20. P

    Equation involving inverse trigonometric function

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  21. ergospherical

    Python Speed of function vs lambda calls in python

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  22. H

    I Symmetry and two electron wave function

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  23. M

    Showing piece-wise function continuous

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  24. M

    Using continuity to evaluate a limit of a composite function

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  25. M

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  26. Euge

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  27. chwala

    Write the given hyperbolic function as simply as possible

    My take; ##2\cosh x = e^x +e^{-x}## I noted that i could multiply both sides by ##e^x## i.e ##e^x⋅2\cosh x = e^x(e^x +e^{-x})## ##e^x⋅2\cosh x = e^{2x}+1## thus, ##\dfrac{e^x}{1+e^{2x}}=\dfrac{\cosh x + \sinh x}{e^x⋅2\cosh x}## ##= \dfrac{\cosh x +...
  28. C

    Is ##f(x)=2^{x}-1## considered an exponential function?

    I wonder if it's ##f(x)=2^{x}-1## considered an exponential function because in my textbook it's stated that the set of values of an exponential function is a set of positive real numbers, while when graphing this function I get values(y line) that are not positive(graph in attachments), so I am...
  29. ananonanunes

    Find limit of multi variable function

    This is what I did: $$\lim_ {(x,y) \rightarrow (1,0)} {\frac {g(x)(x-1)^2y}{2(x-1)^4+y^2}}=\lim_ {(x,y) \rightarrow (1,0)} {g(x)y\frac {(x-1)^2}{2(x-1)^4+y^2}}$$ I know that ##\lim_ {(x,y) \rightarrow (1,0)} {g(x)y}=0## and that ##\frac {(x-1)^2}{2(x-1)^4+y^2}## is limited because ##0\leq...
  30. M

    Computing the derivative of an exponential function

  31. Like Tony Stark

    Mixed states and total wave function for three-Fermion-systems

    I've already calculated the total spin of the system in the addition basis: ##\ket{1 \frac{3}{2} \frac{3}{2}}; \ket{1 \frac{3}{2} \frac{-3}{2}}; \ket{1 \frac{3}{2} \frac{1}{2}}; \ket{1 \frac{3}{2} \frac{-3}{2}}; \ket{0 \frac{1}{2} \frac{1}{2}}; \ket{0 \frac{1}{2} \frac{-1}{2}}; \ket{1...
  32. G

    I Independence of Trace-Partition function

    I am trying to calculate the partition function of the system of two completely decoupled systems. Probability-wise, the decoupled nature means that the PDF is the product of the PDF of each subsystem. I just wanted to be sure that it would translate into: $$ H = \sum_{k_i...
  33. R

    Expressing Feynman Green's function as a 4-momentum integral

    I am a bit confused on how we can just say that (z',p) form a 4-vector. In my head, four vectors are sacred objects that are Lorentz covariant, but now we introduced some new variable and say it forms a 4-vector with momentum. I understand that these are just integration variables but I still do...
  34. M

    B How do I invert this exponential function?

    Preface: I have not done serious math in years. Today I tried to do something fancy for a game mechanic I'm designing. I've got an item with a variable power level. It uses x amount of ammo to produce f(x) amount of kaboom. Initially it was linear, e.g. fL(x) = x, but I didn't like the scaling...
  35. K

    I Best fit to an oscillating function

    Hello! I have a plot of a function, obtained numerically, that looks like the red curve in the attached figure. It is hard to tell, but if you zoom in enough, inside the red shaded area you actually have oscillations at a very high frequency, ##\omega_0##. On top of that you have some sort of...
  36. B

    I Limit as a function, not a value

    Is it possible for a limit of a range of functions to return a function? Example: f(z)= limit (as p approaches 0) (xp-1)/p.
  37. M

    Finding the domain of a composite function

    For this problem, The solution is, However, I tried solving this problem by using the definition of composite function ##f(g(x)) = f(\frac{4}{3x -2}) = \frac{5}{\frac{4}{3x - 2} - 1} = \frac{5}{\frac{6 - 3x}{3x - 2}} = \frac {15x - 10}{6 - 3x}## which only gives a domain ##x ≠ 2##. Would some...
  38. E

    I Integration of Bessel function products (J_1(x)^2/xdx)

    Hello, While reading Sakurai (scattering theory/Eikonal approximation section), I encountered a referenced integral ## \int_0^\infty J_1(x)^2\frac{dx}{x}=1/2 ## I also see this integral from a few places (wolfram, DLMF, etc), so I tried to prove this from various angles (recurrence relations...
  39. S

    I Geometry of series terms of the Riemann Zeta Function

    This is an Argand diagram showing the first 40,000 terms of the series form of the Riemann Zeta function, for the argument ##\sigma + i t = 1/2 + 62854.13 \thinspace i## The blue lines are the first 100 (or so) terms, and the rest of the terms are in red. The plot shows a kind of approximate...
  40. E

    I Determine ## \beta ## as a function of ##\theta## linkage

    I 've been trying find ##\beta## as a function of ##\theta## for this linkage. It's quite the trigonometric mess. Start with the Law of Sines: $$ \frac{\sin \beta}{x} = \frac{\sin \varphi}{R} \implies \boxed{ x = R \frac{\sin \beta}{\sin \varphi} \tag{1} }$$ Relating angles: $$ \theta +...
  41. M

    Limit of a rational function with a constant c

    For this problem, Did they get ## x## approaches one is equivalent to ##t## approaches zero because ##t ∝ (x)^{1/3} + 1##? Many thanks!
  42. O

    Symbolic integration of a Bessel function with a complex argument

    Hello all I am trying to solve the following integral with Mathematica and I'm having some issues with it. where Jo is a Bessel Function of first kind and order 0. Notice that k is a complex number given by Where delta is a coefficient. Due to the complex arguments I'm integrating the...
  43. C

    I How to calculate the Fermi surface nesting function

  44. vinicius_linhares

    Analysis of the Ground Function: f(x) with $$f''(\bar{x})=0$$

    If f(x) is the function of the "ground": My first assumption is that in a certain $$\bar{x}$$, $$f''(\bar{x})=0$$, and from that point I will analyse the situation. The object has initial energy $$E_0=\frac{mv^2}{2}+mgf(x),$$ then $$v=\sqrt{\frac{2}{m}}\sqrt{E_0-mgf(x)}.$$ In each point the...
  45. S

    Trying to reconcile function composition problems with sets & formulas

    I know how to solve each of those problems. For the set one, I look at the output of the S and try to match it with the input of T and then take the pair (input_of_S, output_of_T), and I do that for each pair. As for the formula one, I just plug in x = g(y). My confusion lies in trying to...
  46. F

    Looking for a particular function

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  47. S

    I Load on leaf blower as function of outlet shape

    Would a leaf blower see a different load (in terms of back pressure and flow) depending on whether the outlet tube ends in, say a 1" nozzle, versus a flared out horn of 6" diameter? I am not thinking of viscosity effects here, but rather of Bernoulli type considerations. In the case of the...
  48. E

    A Ballentine on the "multicomponent state function"

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  49. Euge

    POTW Definite Integral of a Rational Function

    Evaluate the definite integral $$\int_0^\infty \frac{x^2 + 1}{x^4 + 1}\, dx$$
  50. abdulbadii

    Failed function within SMPS giving weak current

    What is the very common culprit component and what is its function within SMPS having weak current, far weaker than its rating while its voltage is always perfect?
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