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. M

    A Stream functions and flow around sphere/cylinder

    Hi PF! I am wondering why we define velocity for polar coordinates as $$\vec{V} = \nabla \times \frac{\psi(r,z)}{r} \vec{e_\theta}$$ and why we define velocity in spherical coordinates as $$\vec{V} = \nabla \times \frac{\psi(r,\phi)}{r \sin \phi} \vec{e_\theta}$$ The only thing I don't...
  2. K

    Tangent to Hyperbolic functions graph

    Homework Statement Show that the tangent to ##x^2-y^2=1## at points ##x_1=\cosh (u)## and ##y_1=\sinh(u)## cuts the x-axis at ##{\rm sech(u)}## and the y-axis at ##{\rm -csch(u)}##. Homework Equations Hyperbolic sine: ##\sinh (u)=\frac{1}{2}(e^u-e^{-u})## Hyperbolic...
  3. K

    I An identity of hyperbolic functions

    Prove: ##(\cosh(x)+\sinh(x))^n=\cosh(nx)+\sinh(nx)## Newton's binomial: ##(a+b)^n=C^0_n a^n+C^1_n a^{n-1}b+...+C^n_n b^n## and: ##(a-b)^n~\rightarrow~(-1)^kC^k_n## I ignore the coefficients. $$(\cosh(x)+\sinh(x))^n=\cosh^n(x)+\cosh^{n-1}\sinh(x)+...+\sinh^n(x)$$...
  4. U

    MHB Find the derivative using implicit differentiation (with inverse trig functions)

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  5. C

    Implementing boolean functions with decoder and external gate

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  6. RicardoMP

    I Square integrable wave functions vanishing at infinity

    Hi! For the probability interpretation of wave functions to work, the latter have to be square integrable and therefore, they vanish at infinity. I'm reading Gasiorowicz's Quantum Physics and, as you can see in the attached image of the page, he works his way to find the momentum operator. My...
  7. mastrofoffi

    Show orthogonality of vector-valued functions

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  8. N

    I Green's Function: Hamiltonian and Density of States Explained

    On wikipedia it says the following, "...the Green's function of the Hamiltonian is a key concept with important links to the concept of density of states." https://en.wikipedia.org/wiki/Green%27s_function Can anyone explain why?
  9. B

    How do I find the convolution of two functions with different domains?

    Homework Statement I have the two functions below and have to find the convolution \beta * L Homework Equations Assume a<1 \beta(x)=\begin{cases} \frac{\pi}{4a}\cos\left(\frac{\pi x}{2a}\right) & \left|x\right|<a\\ 0 & \left|x\right|\geq a \end{cases} L(x)=\begin{cases} 1 &...
  10. S

    A Correlation functions in position and momentum space

    What is the relation between the correlators ##\langle 0 | T\phi(x_{1})\phi(x_{2}) | 0 \rangle## and ##\langle 0 | T\phi(p)\phi(x=0) | 0 \rangle##? I can derive the momentum space Feynman rules for ##\langle 0 | T\phi(x_{1})\phi(x_{2}) | 0 \rangle##. Are the momentum space Feynman rules for...
  11. T

    I Equations for functions in the complex domain

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  12. karush

    MHB 10) AP Calculus linear functions

    $\textbf{10)} \\ f(x)\text{ is continuous at all } \textit{x} \\ \displaystyle f(0)=2, \, f'(0)=-3,\, f''(0)=0 $ $\text{let} \textbf{ g } \text{be a function whose derivative is given by}\\ \displaystyle g'(x)=e^{-2 x} (3f(x))+2f'(x) \text{ for all x}\\$ $\text{a) write an equation of the...
  13. X

    I Vector Functions: v(r) Explained

    Is v(r) ≡ v(x,y,z)
  14. DoobleD

    I Components of functions in vector spaces

    I have some conceptual issues with functions in vectors spaces. I don't really get what are really the components of the vector / function. When we look at the inner product, it's very similar to dot product, as if each value of a function was a component : So I tend to think to f(t) as the...
  15. Saracen Rue

    B Proving these two functions intersect at 'a'

    Through experimental observations, I have found that the two functions ##f\left(x\right)=x^{a\left(a-x^3\right)}-a## and ##g\left(x\right)=a^{x\left(x-a^3\right)}-x## will always intersect at ##a## when ##x>0##. Is there a way to mathematically prove this? For instance, simultaneously solving...
  16. toforfiltum

    Finding functions that are a level set and a graph

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  17. K

    I Integral of a special trigonometic functions

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  18. Battlemage!

    I Lim f(x)/g(x) as x->∞ and relative growth rate of functions

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  19. M

    I Difficulty with distribution functions

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  20. S

    Charge densities and delta functions

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  21. M

    Linear transformations, images for continuous functions

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  22. S

    How to make functions right-continuous

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  23. pairofstrings

    Visualize this type of Combined Trigonometric Functions

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  24. J

    Physical difference between various wave functions

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  25. ItsAnshumaan

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  26. beyondlight

    Determine all primitive functions

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  27. Nader AbdlGhani

    B Difference between these functions .

    What's the difference between f(x)=3 and f(x)=3x^0 ? and why Limit of the second function when x\rightarrow0 exists ? and is the second function continuous at x=0 ?
  28. evinda

    MHB Difference of two types of recursive functions

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  29. J

    MHB Prove 1 - 1: Prove Functions are 1 - 1

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  30. Joppy

    MHB Fourier Transform of Periodic Functions

    A tad embarrassed to ask, but I've been going in circles for a while! Maybe i'll rubber duck myself out of it. If f(t) = f(t+T) then we can find the Fourier transform of f(t) through a sequence of delta functions located at the harmonics of the fundamental frequency modulated by the Fourier...
  31. M

    Finding the Inverse Function of f(x) = 1−3x−2x^2 on Domain [-2, -1]

    Homework Statement Let f(x) = 1−3x−2x^2 , x ∈ [−2, −1]. Use the Horizontal Line Test to show that f is 1–1 (on its given domain), and find the range R of f. Then find an expression for the inverse function f −1 : R → [−2, −1]. The Attempt at a Solution I have already done the horizontal line...
  32. E

    A Commutator of field operator with arbitrary functions

    In QFT, the commutation relation for the field operator \hat{\phi} and conjugate momentum is [\phi(x,t),\pi(y,t)] = i\delta(x-y) Maybe this is obvious, but what would the commutator of \phi or \pi and, say, e^{i k\cdot x} be?
  33. Y

    A Dyson's equation and Green's functions

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  34. JulienB

    I Limits of multivariable functions (uniform convergence)

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  35. Evangeline101

    Sinusoidal Functions: describe transformations, sketch graph

    Homework Statement Homework Equations none The Attempt at a Solution -amplitude is 3 -period is 180° -right 60° -down 1 Rough sketch of graph: I would like to know if the graph looks right, is there any improvements to be made? Thanks :)
  36. Evangeline101

    Astrolabe Roadstead tides - Sinusoidal Functions

    Homework Statement Homework Equations 3. The Attempt at a Solution a) The height of the high tide is 4.5 m b) The height of the low tide is 0.25 m c) Period = 12.5 hours k= 360/12.5 = 28.8 amplitude = 2.125 m vertical shift = 2.375 m phase shift = it doesn't look like there is any...
  37. P

    Recurrence relation for Bessel Functions

    Homework Statement I want to prove this relation ##J_{n-1}(x) + J_{n+1}(x)=\frac{2n}{x}J_{n}(x))## from the generating function. The same question was asked in this page with solution. http://www.edaboard.com/thread47250.html My problem is the part with comparing the coefficient. I don't...
  38. M

    Help with Trig Function: Sec(2x)csc(x)sin(2x) and C=cosx

    Homework Statement Let C=cosx. Write sec(2x)csc(x)sin(2x) as a function of C. The Attempt at a Solution Am I on the right track 1/cos(2x) * 1/sin(x) * 2sin(x)cos(x) 1/(cos^2(x)-sin^2(x)) * `1/(sqrt(1-cos^2(x)) * 2(sqrt(1-cos^2(x))cos(x) What would i do from here?
  39. Evangeline101

    Sinusoidal Functions: Niagara Falls Skywheel....

    Homework Statement Homework Equations The Attempt at a Solution a) Here is a sketch of the graph. The lowest point on the Ferries Wheel is 2.5 m and the highest point is 2.5 m + 50.5 m = 53 m. It completes a full cycle every 120 seconds and starts at the lowest point. b) The highest...
  40. dreens

    I Orthogonal 3D Basis Functions in Spherical Coordinates

    I'd like to expand a 3D scalar function I'm working with, ##f(r,\theta,\phi)##, in an orthogonal spherical 3D basis set. For the angular component I intend to use spherical harmonics, but what should I do for the radial direction? Close to zero, ##f(r)\propto r##, and above a fuzzy threshold...
  41. M

    Find the Domain and Range of Functions with Given Domain and Range Values

    Homework Statement Suppose f is a function with domain [-2,10] and range [5,10]. Find the domain and range of the following functions. (a) f(2x+4) (b) 2f(x)+4 The Attempt at a Solution [/B] Would I just substitute the in the domain and range values to find the answer?
  42. M

    I Relations & Functions: Types, Examples, Homomorphism

    Hello every one . A relation ( is a subset of the cartesian product between Xand Y) in math between two sets has spatial types 1-left unique ( injective) 2- right unique ( functional ) 3- left total 4- right total (surjective) May question is 1- a function ( map...
  43. A

    Use of floor and ceiling functions in physics problems

    Homework Statement explained on document attached Homework Equations Energy on a spring and work done by friction The Attempt at a Solution Included on document https://docs.google.com/document/d/1FNrmIkkWzyZJNsbGbq_DYMyMZbpyMcAYKYk9iTbdR-4/edit?usp=sharing
  44. K

    Properties of Wave Functions and their Derivatives

    Homework Statement I am unsure if the first statement below is true. Homework Equations \frac{\partial \psi^*}{\partial x} \frac{\partial^2 \psi}{\partial x^2}=\frac{\partial^2 \psi}{\partial x^2}\frac{\partial \psi^*}{\partial x} Assuming this was true, I showed that \int \frac{\partial...
  45. orion

    I Understanding the Transition Functions for S^1 Using Atlas Charts

    I am confused about the procedure for finding the transition functions given an atlas. I understand the theory; it's applying it to real life examples where I have my problem. So for example, take S1 (the circle). I want to use 2 charts given by: U1 = {α: 0 < α < 2π} φ1 = (cos α, sin α) U2...
  46. orion

    I Boundedness and continuous functions

    I am working my way through elementary topology, and I have thought up a theorem that I am having trouble proving so any help would be greatly appreciated. ---------------------- Theorem: Let A ⊂ ℝn and B ⊂ ℝm and let f: A → B be continuous and surjective. If A is bounded then B is bounded...
  47. Rectifier

    Understanding Limits of Composed Functions at Infinity

    The problem $$ \lim_{x \rightarrow \infty} \frac{(\ln x)^{300}}{x} $$ The attempt ## \lim_{x \rightarrow \infty} (\ln x)^{300} = \infty## since ## \lim_{x \rightarrow \infty} f(x) = A## and ## \lim_{x \rightarrow \infty} g(x) = \infty ## thus ## \lim_{x \rightarrow \infty}f(g(x)) = A ##. ##...
  48. E

    Units of constants in transfer functions?

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  49. J

    I Procedurally generated polynomial functions

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  50. awholenumber

    I Question ,trigonometric identities equation and functions ?

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