What is Functions: Definition and 1000 Discussions

In mathematics, the trigonometric functions (also called circular functions, angle functions or goniometric functions) are real functions which relate an angle of a right-angled triangle to ratios of two side lengths. They are widely used in all sciences that are related to geometry, such as navigation, solid mechanics, celestial mechanics, geodesy, and many others. They are among the simplest periodic functions, and as such are also widely used for studying periodic phenomena through Fourier analysis.
The trigonometric functions most widely used in modern mathematics are the sine, the cosine, and the tangent. Their reciprocals are respectively the cosecant, the secant, and the cotangent, which are less used. Each of these six trigonometric functions has a corresponding inverse function (called inverse trigonometric function), and an equivalent in the hyperbolic functions as well.The oldest definitions of trigonometric functions, related to right-angle triangles, define them only for acute angles. To extend these definitions to functions whose domain is the whole projectively extended real line, geometrical definitions using the standard unit circle (i.e., a circle with radius 1 unit) are often used. Modern definitions express trigonometric functions as infinite series or as solutions of differential equations. This allows extending the domain of sine and cosine functions to the whole complex plane, and the domain of the other trigonometric functions to the complex plane from which some isolated points are removed.

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  1. L

    A Derivatives of functions in ODE

    For ordinary differential equation y''(x)+V(x)y(x)+const y(x)=0 for which ##\lim_{x \to \pm \infty}=0## if we have that in some point ##x_0## the following statement is true ##y(x_0)=y'(x_0)=0## is then function ##y(x)=0## everywhere?
  2. lep11

    Prove functions f and g are continuous in the reals

    Homework Statement Prove functions f and g are continuous in ℝ. It's known that: i) lim g(x)=1, when x approaches 0 ii)g(x-y)=g(x)g(y)+f(x)f(y) iii)f2(x)+g2(x)=1 The Attempt at a Solution [/B] g(0) has to be equal to 1 because it's known that lim g(x)=1, when x approaches 0. Otherwise g won't...
  3. E

    I Eigenspectra and Empirical Orthogonal Functions

    Are the Eigenspectra (a spectrum of eigenvalues) and the Empirical Orthogonal Functions (EOFs) the same? I have known that both can be calculated through the Singular Value Decomposition (SVD) method. Thank you in advance.
  4. Dopplershift

    I Determining the Rate at Which Functions approach Infinity

    With basic fractions, the limits of 1/x as x approaches infinity or zero is easily determine: For example, \begin{equation} \lim_{x\to\infty} \frac{1}{x} = 0 \end{equation} \begin{equation} \lim_{x\to 0} \frac{1}{x} = \infty \end{equation} But, we with a operation like ##\frac{f(x)}{g(x)}##...
  5. M

    B Continuous and differentiable functions

    "If a function can be differentiated, it is a continuous function" By contraposition: "If a function is not continuous, it cannot be differentiated" Here comes the question: Is the following statement true? "If a function is not right(left) continuous in a certain point a, then the function...
  6. ELB27

    Product of a delta function and functions of its arguments

    Homework Statement I am trying to determine whether $$f(x)g(x')\delta (x-x') = f(x)g(x)\delta (x-x') = f(x')g(x')\delta(x-x')$$ where \delta(x-x') is the Dirac delta function and f,g are some arbitrary (reasonably nice?) functions. Homework Equations The defining equation of a delta function...
  7. kenyanchemist

    I Demerits radial distribution functions

    i have a question, why is the plot of r2(Ψ2p)2 not a good representation of the probability of finding an electron at a distance r from the nucleus in a 2p orbital
  8. V

    Prove the integral is in the range of f

    Homework Statement If f: [0,1] \rightarrow \mathbb{R} is continuous, show that (n+1) \int_0^1 x^n f(x) \mathrm{d}x is in the range of f Homework Equations (n+1) \int_0^1 x^n f(x) \mathrm{d}x=\int_0^1 (x^{n+1})' f(x) \mathrm{d}x The Attempt at a Solution I tried integration by parts, but that...
  9. E

    A Functions with "antisymmetric partial"

    Sorry for the terribly vague title; I just can't think of a better name for the thread. I'm interested in functions ##f:[0,1]^2\to\mathbb{R}## which solve the DE, ##\tfrac{\partial}{\partial y} f(y, x) = -\tfrac{\partial}{\partial x} f(x,y) ##. I know this is a huge collection of functions...
  10. S

    MHB Anatomy of piece-wise functions

    Hi! I'm looking at some piece-wise function right now and I can't help but wonder what all these parts are called. I'm learning to use and write this type of functions now and I think I have a pretty good understanding of how they work. I even took the extra step of learning some "set builder...
  11. Drakkith

    I Defining Functions as Sums of Series

    My Calculus 2 teacher's lecture slides say: Many of the functions that arise in mathematical physics and chemistry, such as Bessel functions, are defined as sums of series. I was just wondering how this was different from the basic functions that we've already worked with. Are they not...
  12. A

    Finding the Domain of a Trigonometric Function

    Homework Statement Find the domain of this function and check with your graphing calculator: f(x)=(1+cosx)/(1-cos2x) Homework EquationsThe Attempt at a Solution i get to (1+cosx)/(1+cosx)(1-cosx) which is factored. so then setting each one to zero one at a time i figure out that cosx = -1 and...
  13. S

    I Triplet States and Wave Functions

    Why is the triplet state space wave function ΨT1=[1σ*(r1)1σ(r2)-1σ(r1)1σ*(r2)] (ie. subtractive)? How does it relate to its antisymmetric nature? Also, why is this opposite for the spin wave function α(1)β(2)+β(1)α(2) (ie. additive)? And why is this one symmetric even though it describes the...
  14. Geologist180

    What Is g'(2) for the Function G = (1/f^-1)?

    Homework Statement Suppose that f has an inverse and f(-4)=2, f '(-4)=2/5. If G= (1/f-1) what is g '(2) ? If it helps the answer is (-5/32) Homework Equations [/B] f-1'(b)=1/(f')(a) The Attempt at a Solution Im not really sure how to start this problem. I am familiar with how to use the...
  15. Alanay

    How do I calculate inverse trig functions?

    On the paper I'm reading the arctan of 35 over 65 is approx. 28.30degrees. When I use the Google calculator "arctan(35/65)" gives me 0.493941369 rad. What am I doing wrong?
  16. B

    MHB Another field lines of 3D vector functions question

    I am trying to find the field lines of the 3D vector function F(x, y, z) = yi − xj +k. I began by finding dx/dt =y, dy/dt = -x, dz/dt = 1. From here I computed dy/dx = -x/y, and hence y^2 + x^2 = c. For dz/dt = 1, I found that z = t + C, where C is a constant. I am unsure where to go from...
  17. B

    MHB Field lines of 3D vector functions

    My question regards finding the field lines of the 3D vector function F(x,y,z) = yzi + zxj + xyk. I was able to compute them to be at the curves x^2 - y^2 = C and x^2 -z^2 = D, where C and D are constants. From my understanding the field lines will occur at the intersection of these two...
  18. DaTario

    I Examples of Basic Potential functions

    Hi All, In teaching the basics of quantum mechanics, one has often to introduce potential functions such as the step, the barrier and the well. When I try to get some example of the physical environment of a particle that could correspond to a step function, for instance, what comes out is...
  19. A

    B Monotony of composite functions

    So, it is known and easy to prove that if you have f : D -> G and g : G -> B then -if both f and g have the same monotony => fοg is increasing -if f and g have different monotony => fοg is decreasing But the reciprocal of this is not always true (easy to prove with a contradicting example)...
  20. Raptor112

    A Quantum Optics Question and Wigner Functions

    I understand that Wigner function is a quasi-probability distibution as it can take negative values, but in quantum optics I see that the Q function is mentioned as often. So what is the difference between the Q function and the Wigner Function?
  21. squelch

    How many surjective functions are there from {1,2,...,n} to {a,b,c,d}?

    Homework Statement Count the number of surjective functions from {1,2,...,n} to {a,b,c,d}. Use a formula derived from the following four-set venn diagram: Homework Equations None provided. The Attempt at a Solution First, I divided the Venn diagram into sets A,B,C,D and tried to express...
  22. G

    Finding the Area Bounded by Two Functions

    Homework Statement Find area bounded by parabola y^2=2px,p\in\mathbb R and normal to parabola that closes an angle \alpha=\frac{3\pi}{4} with the positive Ox axis. Homework Equations -Area -Integration -Analytic geometry The Attempt at a Solution For p>0 we can find the normal to parabola...
  23. erbilsilik

    What are the expansions of Bose functions for studying thermodynamic behavior?

    Homework Statement To study the thermodynamic behavior of the limit $$z\rightarrow1$$ it is useful to get the expansions of $$g_{0}\left( z\right),g_{1}\left( z\right),g_{2}\left( z\right)$$ $$\alpha =-\ln z$$ which is small positive number. From, BE integral, $$g_{1}\left( \alpha \right)...
  24. BubblesAreUs

    Python problem: Plotting two functions against each other

    Homework Statement Enter a minimum height and velocity into plot function and return a velocity-height plot. Homework EquationsThe Attempt at a Solution # Find length of general list n = len(K) # Build a list for time [0,20] seconds ( Global) time = n*[0.0] # Acceleration of gravity g =...
  25. O

    Curve fitting (Linearization) of functions (and thus graphs)

    Ok, first week of first year of undergraduate physics lab and they explain that we want all our graphs to be linear, and in order to do that we can change our x and y axes to be log(x) or y^2 or whatever. They did some simple examples such as y=(k/x)+c and explained that if the x axes is 1/x we...
  26. Math Amateur

    MHB Distributing the Product of Functions over Composition of Functions

    I am reading John M. Lee's book: Introduction to Smooth Manifolds ... I am focused on Chapter 3: Tangent Vectors ... I need some help in fully understanding Lee's definition and conversation on pushforwards of F at p ... ... (see Lee's conversation/discussion posted below ... ... ) Although...
  27. I

    Integrating Implicit Functions

    In one of the homework sheets my teacher gave us, we had to calculate area geometrically (meaning no integration was used). Some parts, she said, we needed to just eyeball which I hate doing. In this case the top left portion of a circle described by the equation...
  28. Titan97

    Number of functions such that f(i) not equal to i

    Homework Statement ##A=\{1,2,3,4,5\}##, ##B=\{0,1,2,3,4,5\}##. Find the number of one-one functions ##f:A\rightarrow B## such that ##f(i)\neq i## and ##f(1)\neq 0\text{ or } 1##. Homework Equations Number of derangements of n things =...
  29. Kingyou123

    Which functions are missing from {1,2,3} to {a,b} and why?

    Homework Statement How many functions are there from {1,2,3} to {a,b}? Which are injective? Which are surjective? Homework Equations n^m. gives the number of functions The Attempt at a Solution To me the number of functions that can be made are 6 because 3x2=6 but I have read online that n^m...
  30. 1

    Differentiability of piece-wise functions

    Hello, Me and my friend were talking about differentiability of some piece-wise functions, but we thought of a problem that we could were not able to come to an agreement on. If the function is: y=sin(x) for x≠0 and y=x^2 for x=0, Is this function differentiable? The graph looks like a normal...
  31. F

    Prove that three functions form a dual basis

    Homework Statement Homework Equations[/B] The Attempt at a Solution From that point, I don't know what to do. How do I prove linear independence if I have no numerical values? Thank you.
  32. ognik

    How Do Function Widths and Uncertainty Principles Relate in Quantum Mechanics?

    Homework Statement ## \phi(k_x) = \begin{cases}\phantom{-} \sqrt{2 \pi},\; \bar{k_x} - \frac{\delta}{2} \le k_x \le \bar{k_x} + \frac{\delta}{2} \\ - \sqrt{2 \pi},\; \bar{k_x} - \delta \le k_x \le \bar{k_x} - \frac{\delta}{2} \:AND \: \bar{k_x} + \frac{\delta}{2} \le k_x \le \bar{k_x} +...
  33. A

    Bessel functions and the dirac delta

    Homework Statement Find the scalar product of diracs delta function ##\delta(\bar{x})## and the bessel function ##J_0## in polar coordinates. I need to do this since I want the orthogonal projection of some function onto the Bessel function and this is a key step towards that solution. I only...
  34. ognik

    How Do Widths of Functions Relate to Uncertainty Principles?

    (corrections edited in) 1. Homework Statement Assume ## \psi(x, 0) = e^{-\lambda |x|} \: for \: -\infty < x < +\infty ##. Calculate ## \phi(k_x) ## and show that the widths of ## \phi, \psi ##, reasonably defined, satisfy ##\Delta x \Delta k_x \approx 1 ## Homework Equations ## \phi(k_x) =...
  35. A

    Question about multiple functions for a first order ODE

    The question is as follows: Suppose you find an implicit solution y(t) to a first order ODE by finding a function H(y, t) such that H(y(t), t) = 0 for all t in the domain. Suppose your friend tries to solve the same ODE and comes up with a different function F(y, t) such that F(y(t), t) = 0 for...
  36. W

    "Interesting" or general Mathematical User-defined Functions

    Hi all, just curious. I am just learning about user-defined functions in MSSQL2014. What kind of Math can we do with it? Didn't get much useful from my search.
  37. ORF

    C/C++ How to read a binary file using C++11 functions?

    Hello I'm using the C functions for reading binary files: #include <iostream> #include <stdio.h> void main(){ /*********/ uint32_t head=0; FILE *fin = NULL; fin = fopen("myFile.bin","r"); while(myCondition){ fread(&head,4,1,fin); std::cout << std::hex << head...
  38. Math Amateur

    MHB Real Valued Functions on R^3 - Chain Rule ....?

    I am reading Barrett O'Neil's book: Elementary Differential Geometry ... I need help to get started on Exercise 4(a) of Section 1.1 Euclidean Space ... Exercise 4 of Section 1.1 reads as follows:Can anyone help me to get started on Exercise 4(a) ... I would guess that we need the chain rule...
  39. YogiBear

    Mechanical variation involving auxiliary functions

    Homework Statement A chain of length L and uniform mass per unit length ρ is suspended in a uniform gravitational field. The potential energy U[y] and length l[y] functionals of the chain can be written in terms of y(x) as follows: U[y] = ρg*Int(y(1+y'^2)^1/2 dx) l[y] = Int((1+y'^2)^1/2)...
  40. M

    Chain Rule W/ Composite Functions

    Homework Statement If d/dx(f(x)) = g(x) and d/dx(g(x)) = f(x2), then d2/dx2(f(x3)) = a) f(x6) b) g(x3) c) 3x2*g(x3) d) 9x4*f(x6) + 6x*g(x3) e) f(x6) + g(x3) Homework EquationsThe Attempt at a Solution The answer is D. Since d/dx(f(x)) = g(x), I said that d/dx(f(x3)) should equal 3x2*g(x3), then...
  41. Ssnow

    Moment maps and Morse functions

    It is know that let ##M## a compact symplectic manifold with ##G=T^{d_{T}}## a torus of dimension ##d_{T}## acting on ##M## in Hamiltonian fashion with Moment map ##\Phi:M\rightarrow \mathfrak{t}^{*}##, then ##\Phi^{\xi}=\langle \Phi(m),\xi\rangle## is a Morse function in each of its component...
  42. Icaro Lorran

    Envelope of a parametric family of functions

    Consider the map ##\phi (t,s) \mapsto (f(t,s),g(t,s))##, a point belonging to the envelope of this map satisfy the condition ##J_{\phi}(t,s)=0##. What is the role of the Jacobian in maps like these and why points in the envelope have to satisfy ##J_{\phi}(t,s)=0##?
  43. D

    Factorization of floor functions of fractions

    hey so if you are taking a floor function of a fraction >1, is there any way to predict anything about it's factorization? what about when the numerator is a factorial and the denominator is made up of factors that divide said factorial but to larger exponents then those that divide the...
  44. F

    Odd/Even functions and integration of them

    I was not sure where to post this here or in calculus, but seeing as the underlying basic principle of my question is regarding parity of functions I am posting it here, but feel free to move if needed. Basically I am getting ready for a (intro to) QM exam and I still struggle with some basic...
  45. G

    MHB Inverse trigonometric functions

    What's $1. ~ \displaystyle \arccos(\cos\frac{4\pi}{3})?$ Is this correct? The range is $[0, \pi]$ so I need to write $\cos\frac{4\pi}{3}$ as $\cos{t}$ where $t$ is in $[0, \pi]$ $\cos(\frac{4\pi}{3}) = \cos(2\pi-\frac{3\pi}{3}) = \cos(\frac{2\pi}{3}) $ so the answer is $\frac{2\pi}{3}$
  46. A

    Integral equivalent to fitting a curve to a sum of functions

    Hello, I am searching for some kind of transform if it is possible, similar to a Fourier transform, but for an arbitrary function. Sort of an inverse convolution but with a kernel that varies in each point. Or, like I say in the title of this topic a sort of continuous equivalent of fitting a...
  47. F

    Functions and their return....

    Hello Forum, I am trying to get clear on the return statement when defining functions in C. A function is a group of statements that together perform a certain task. A function usually receives some input arguments which it uses to produce some output arguments. In C, we must specify what type...
  48. N

    Is a Line Intersecting at One Point a 1-1 Function?

    if we draw a line parallel to the x- axis and passes through a point in the image and the graph intersects at one point is this a one to one function ?
  49. N

    A question on plotting functions on a graph

    when i was reading a supplementary notes doc from open course ware fro MIT on single variable calculus there was a description about a graphical representation of a single valued function as " if each line parallel to the y- axis and which passes through a point in the domain intersects the...
  50. N

    A question on plotting functions on a graph

    when i was reading a supplementary notes doc from open course ware fro MIT on single variable calculus there was a description about a graphical representation of a single valued function as " if each line parallel to the y- axis and which passes through a point in the domain intersects the...
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