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. kostoglotov

    How transfer functions become Bode plots

    Homework Statement I have a question related to taking the logs of transfer functions, getting the individual Bode plots of each subsequent factor, and adding those plots graphically. I'm working from Fundamentals of Electric Circuits, 5th edt. Let me start with the following screen capture...
  2. U

    I How to solve this system of equations of trig functions

    I've written it out and it seems impossible. I get -50(sin^2(alpha)) = 86.63 cos(alpha) sin(alpha) - 6.54. Where would I go from there?
  3. N

    MHB Show 2 functions have the same anti-derivative

    So I have to show 2sin^2(x) and -cos(2x) have the same antiderative. Here's how I approached this. 2sin^2(x) = 1-cos2x ==> u = 2x intergral of that is (u - sinu)/2 + c = x - (sinx)/2 + c -cos2x ==> u = 2x intregal of that is (-sinu)/2 + c= -(sin2x)/2 + c Have I calculated/approached this...
  4. T

    Using matrices for functions -- transformations and translation

    Homework Statement Happy new year all. I was wondering if you can use matrices to translate and transform a function? So for example if I were to take the function $$f(x)=x^2+4x$$ and I want to the translate and transform the equation to $$2f(x+4)$$. Can this be done by matrices. I know how...
  5. N

    A Do higher dimensional branes have wave functions?

    Do higher dimensional branes, like the super membrane (which is a 2D brane) or the NS5/M5 brane, have wave functions? I know that they become unstable once they are quantized, but does that mean that they do not have wave functions? You will never here about any thing regarding an M2 wave...
  6. J

    MHB Integral of Fresnel functions

    I have been working on this for several days but getting nowhere. Any help would be great. \begin{align} &\int_0^x dy\,y^2 \cos(y^2) C^2 \!\!\left(\!\frac{\sqrt{2}\,y}{\sqrt\pi}\!\right) \end{align} In reality only the first one is causing me troubles, however I have pasted the entire...
  7. Q

    I Analysis of a general function with a specific argument

    Hello everybody, I'm currently helping a friend on an assignment of his, but we are both stumbled on this exercise, I'm posting it here We define a function ##f## which goes from ##\mathbb{R}## to ##\mathbb{R}## such that its argument maps as $$ x \mapsto...
  8. M

    B Is it impossible to prove that some functions are periodic?

    Heres an example. Let G(s) be the moving average of all previous values of f(s). G(s) and F(s) intersect at multiple points. Is it possible to prove that the intersections happen periodically?
  9. BiGyElLoWhAt

    I A question about boundary conditions in Green's functions

    I have a couple homework questions, and I'm getting caught up in boundary applications. For the first one, I have y'' - 4y' + 3y = f(x) and I need to find the Green's function. I also have the boundary conditions y(x)=y'(0)=0. Is this possible? Wouldn't y(x)=0 be of the form of a solution...
  10. T

    How Do Bessel Functions Predict Sideband Amplitudes in FM Modulation?

    Homework Statement For the FM modulation, the amplitudes of the side bands can be predicted from v(t)=ΣAJn(I)sin(ωt) Where is a sideband frequency and Jn(I) is the Bessel function of the first kind and the nth order evaluated at the modulation index .Given the table of Bessel functions...
  11. J

    Applied Books on complex valued functions and solution of PDE

    Hello folks, 1.- In geometry we study for example the conic sections, their exentricity and properties. I was wondering what part of the mathematical science studies the different properties of complex valued distributions. One example are the spherical armonics. I guess mathematicians have...
  12. Stoney Pete

    I Can an ordered pair have identical elements?

    Hi guys, Here is a wacky question for you: Suppose you have a simple recursive function f(x)=x. Given the fact that a function f(x)=y can be rewritten as a set of ordered pairs (x, y) with x from the domain of f and y from the range of f, it would seem that the function f(x)=x can be written...
  13. BiGyElLoWhAt

    I A somewhat conceptual question about Green's functions

    I just did a problem for a final that required us to use a green's function to solve a diff eq. y'' +y/4 = sin(2x) I went through and solved it and got a really nasty looking thing, but I checked it in wolfram and it works out. Now, my question is this: After I got the solution from my greens...
  14. A

    Substituting functions in limits

    Homework Statement I'm trying hard to understand as my professor hasn't taught(nor does my textbook) on how this works. It is known that $$\lim_{x \to 0}\frac{f(x)}{x} = -\frac12$$ Solve $$\lim_{x \to 1}\frac{f(x^3-1)}{x-1}.$$ Homework EquationsThe Attempt at a Solution OK.. so I do this...
  15. T

    A Are all wave functions with a continuum basis non-normalizable?

    For example, I am following the below proof: Although the above derivation involves a projection on the position basis, it appears one can generalize this result by using any complete basis. So despite it not being explicitly mentioned here, are all wave functions with any continuum basis...
  16. Mr Davis 97

    I Linear independence of functions

    Is there a difference between the linear independence of ##\{x,e^x\}## and ##\{ex,e^x\}##? It can be shown that both only have the trivial solution when represented as a linear combination equal to zero. However, the definition of linear independence is: "Two functions are linearly independent...
  17. PhanicKnight

    How Do You Draw a Function from Its Equation?

    Homework Statement I need to draw this function: however I don't get how? I have the solution but I don't understand how do I get that from the given function. Someone please try to explain? Thanks
  18. A

    Trigonometric functions and integrals

    Homework Statement I'm searching for the integral that gives arcosu Homework Equations as we know : ∫u'/[1-u^2]^0.5 dx = arcsinu derivative of arccosu = -u'/[1-u^2]^0.5 + C derivative of arcsinu= u'/[1-u^2]^0.5 The Attempt at a Solution when I type the -u'/[1-u^2]^0.5 on the online integral...
  19. M

    MHB How can we apply the functions?

    Hey! :o Let $\text{Val} = \{0, 1\}^8$, $\text{Adr} = \{0, 1\}^{32}$ and $\text{Mem} = \text{Val}^{\text{Adr}}$. The addition modulo $2^8$ of two numbers in binary system of length $8$, is given by the mapping: $$\text{add}_{\text{Val}} : \text{Val}\times \text{Val}\rightarrow \text{Val} \\...
  20. V

    A Proof of quantum correlation functions

    Reading through David Tong lecture notes on QFT.On pages 76, he gives a proof on correlation functions . See below link: [QFT notes by Tong][1] [1]: http://www.damtp.cam.ac.uk/user/tong/qft/qft.pdfI am following the proof steps to obtain equation (3.95). But several intermediate steps of the...
  21. J

    Studying Differential equations with complex functions?

    Hi folks, When you have a differential equation and the unknown function is complex, like in the Schrodinger equation, What methods should you use to solve it? I mean, there is a theory of complex functions, Laurent series, Cauchy integrals and so on, I guess if it would be possible to...
  22. E

    Determine the Laplace transform for the following functions

    Homework Statement Can someone check my work? Homework EquationsThe Attempt at a Solution 1. ##\frac{1}{s+2}+\frac{1}{s^2+1}## 2. ##\frac{2}{s}+\frac{3}{s+4}## 3. ##\frac{s*sin(-2)+cos(-2)}{s^2+1}## 4. ##\frac{1}{(s+1)^2}## 5. Don't really know how to do this one...
  23. M

    MHB Can Discontinuity and Non-Derivability Exist in Strongly Concave Functions?

    Hey! :o Could you give me an example of a strong concave function $f:[0,3]\rightarrow \mathbb{R}$ that is not continuous? (Wondering) We have that $f''(x)<0$. Since the function has not to be continuous, the derivatives are neither continuous, are they? (Wondering) Is maybe the...
  24. evinda

    MHB The functions is equal to zero for x=0

    Hello! (Wave) We consider the following Cauchy problem $u_t=u_{xx} \text{ in } (0,T) \times \mathbb{R} \\ u(0,x)=\phi(x) \text{ where } \phi(x)=-\phi(-x), x \in \mathbb{R} $ I want to show that $ u(t,0)=0, \forall t \geq 0 $. We have the following theorem: Let $\phi \in...
  25. Kara386

    Structure functions in electron-proton scattering

    Homework Statement In electron proton scattering, ##\int_0^{1} F^p_2(x)dx = 0.18## For the neutron in electron deuteron scattering, ##\int_0^1F^n_2(x)dx = 0.12## Therefore determine the ratio ##\frac{\int_0^1xu^p_v(x)dx}{\int_0^1xd^p_v(x)dx}##. Homework EquationsThe Attempt at a Solution...
  26. ShayanJ

    A Green functions and n-point correlation functions

    Green functions are defined in mathematics as solutions of inhomogeneous differential equations with a dirac delta as the right hand side and are used for solving such equations with a generic right hand side. But in QFT, n-point correlation functions are also called Green functions. Why is...
  27. S

    A Free theory time-ordered correlation functions with derivatives of fields

    Consider the following time-ordered correlation function: $$\langle 0 | T \{ \phi(x_{1}) \phi(x_{2}) \phi(x) \partial^{\mu}\phi(x) \partial_{\mu}\phi(x) \phi(y) \partial^{\nu}\phi(y) \partial_{\nu}\phi(y) \} | 0 \rangle.$$ The derivatives can be taken out the correlation function to give...
  28. F

    I Correlation functions and correlation length

    I thought I understood the concept of a correlation function, but I having some doubts. What exactly does a correlation function quantify and furthermore, what is a correlation length. As far as I understand, a correlation between two variables ##X## and ##Y## quantifies how much the two...
  29. H

    I Inversion of functions that aren't 1-1

    If ##t## is a function of ##r##, then we may in theory find ##r## as a function of ##t##, as claimed in the last paragraph of the attachment below. My issue is this is only true if ##t## is a 1-1 function of ##r##. Otherwise, suppose ##t=r^2##. Then ##r=\pm\sqrt{t}##, which isn't a function. I...
  30. S

    I Newton's method for approximating solutions of functions

    In my calculus textbook, it shows that a function's solution can be approximated using an approximated function tangent to the original function based on an approximated solution, where the equation to find the approximated is L(x) = f(X0) + f'(X0)*(X-X0), where when rearranged, gives x = Xo -...
  31. dykuma

    Integrating Complex Functions in the Complex Plane

    Homework Statement Evaluate the following line integrals in the complex plane by direct integration. Homework Equations Z= x+i y = Cos(θ) +i Sin(θ) = e^i*θ The Attempt at a Solution I'm not sure how to evaluated this by hand. I tried using Z= x+i y = Cos(θ) +i Sin(θ), and evaluating the...
  32. T

    Transfer functions of active filters with Amplification

    Homework Statement Derive the transfer function for both circuits \frac{V_{out}}{V_{in}} sketch Bode plots for each circuit (amplitude and phase) Homework Equations Z_c=\frac{1}{j{\omega}C}~and~{\omega}_C=\frac{1}{RC} The Attempt at a Solution We can treat this as a potential divider using the...
  33. ShayanJ

    A Differential equations without Green functions

    Are there differential equations that, for some reason, don't have a Green function? Are there conditions for a DE to satisfy so that it can have a Green function? Thanks
  34. L

    A Ortogonality of two variable functions

    In functional analysis functions ##f## and ##g## are orthogonal on the interval ##[a,b]## if \int^b_a f(x)g(x)dx=0 But what if we have functions of two variables ##f(x,y)## and ##g(x,y)## that are orthogonal on the interval ##[a,b]##. Is there some definitions \int^b_a f(x,z)g(z,y)dz=0?
  35. evinda

    MHB Maximum principle for subharmonic functions

    Hello! (Wave) I have a question about the proof of the maximum principle for subharmonic functions. The maximum principle is the following: The subharmonic in $\Omega$ function $v$ does not achieve its maximum at the inner points of $\Omega$ if it is not constant. Proof: We suppose that at...
  36. A

    MHB Find the exact value of each of the remaining trigonometric functions of theta

    sin\theta 3/5
  37. Alvis

    I Complex Analysis Harmonic functions

    Suppose u(x,y) and v(x,y) are harmonic on G and c is an element of R. Prove u(x,y) + cv(x,y) is also harmonic. I tried using the Laplace Equation of Uxx+Uyy=0 I have: du/dx=Ux d^2u/dx^2=Uxx du/dy=Uy d^2u/dy^2=Uyy dv/dx=cVx d^2v/dx^2=cVxx dv/dy=cVy d^2v/dy^2=cVyy I'm not really sure how to...
  38. Erenjaeger

    Which of these relations are functions of x on R

    Mentor note: moved to homework section y = sin(x) y = cos(x) y = tan(x) y = csc(x) y = sec(x) y = cot(x) (a) 0 (b) 4 (c) 6 (d) 2 I thought it was (c) because i graphed all the trig functions and they passed the vertical line test but the answer sheet is saying (d) 2
  39. Mr Davis 97

    Showing that exponential functions are linearly independent

    Homework Statement If ##r_1, r_2, r_3## are distinct real numbers, show that ##e^{r_1t}, e^{r_2t}, e^{r_3t}## are linearly independent. Homework EquationsThe Attempt at a Solution By book starts off by assuming that the functions are linearly dependent, towards contradiction. So ##c_1e^{r_1t}...
  40. S

    A Correlation functions in an interacting theory

    Given the theory $$\mathcal{L}=\frac{1}{2}(\partial_{\mu}\phi)^{2}-\frac{1}{2}m_{\phi}^{2}\phi^{2}+\partial_{\mu}\chi^{*}\partial^{\mu}\chi-m_{\chi}^{2}\chi^{*}\chi+\mathcal{L}_{\text{int}},\qquad \mathcal{L}_{\text{int}}=-g\phi\chi^{*}\chi,$$ the time-correlation function ##\langle \Omega |...
  41. A

    I Are the functions for mixed derivative always equal?

    Hi all, I understand that the mixed partial derivative at some point may not be equal if the such mixed partial derivative is not continuous at the point, but are the actual functions of mixed partial derivatives always equal? In other words, if I simply compute the mixed partial derivatives...
  42. J

    I Explaining Music Notes Consonance with Wave Functions

    Hello all, First of all, I am aware that dissonance and consonance between pitches also depend to an extent by culture and musical origin but there also seems to be some degree of objective perception among people that can be explained scientifically. Also, I'm very new to this so I could be...
  43. V

    Modeling WIth Sinusoidial Functions

    Homework Statement The water depth in a harbor is 21m at high tide and 11m at low tide. Once cycle is completed every 12 hrs. (a) Find equation for the depth as a function of time. (b) Draw a graph for 48 hrs after low tide, which occurred at 14:00. (c) State the times where the water...
  44. C

    Limits of Multivariable Functions

    Homework Statement Find the following limit: Homework EquationsThe Attempt at a Solution My lecturer has said that rational functions which are a ratio of two polynomials are continuous on R^2. He also said that the limits of continuous functions can be computed by direct substitution. The...
  45. LLT71

    I What is the relationship between dot products and orthogonality of functions?

    first of all assume that I don't have proper math knowledge. I came across this idea while I was studying last night so I need to verify if it's valid, true, have sense etc. orthogonality of function is defined like this: https://en.wikipedia.org/wiki/Orthogonal_functions I wanted to...
  46. Kushwoho44

    State Functions for Internal Energy and Enthelphy

    Hi, As is commonly known, u = u(T,v) h = u(T,p) I've worked with some maths proofs of this a while ago, but do you guys have an intuitive way of understanding this without the maths, that is, why the state function for internal energy is defined by intensive volume and enthalpy with pressure...
  47. S

    I Differentiability of multivariable functions

    What does it mean for a ##f(x,y)## to be differentiable at ##(a,b)##? Do I have to somehow show ##f(x,y)-f(a,b)-\nabla f(a,b)\cdot \left( x-a,y-b \right) =0 ##? To show the function is not though, it's enough to show, using the limit definition, that the partial derivative approaching in one...
  48. F

    I Canonical transformations and generating functions

    I've been reading about canonical transformations in Hamiltonian mechanics and I'm a bit confused about the following: The author considers a canonical transformation $$q\quad\rightarrow\quad Q\quad ,\quad p\quad\rightarrow\quad P$$ generated by some function ##G##. He then considers the case...
  49. N

    MHB Finding f/g: Composite Functions

    The questions is asking me to find \frac{f}{g} basically , the question is asking me to find the answer , even though i know it, i can't get my head around it. the composite function is f(x)=x^2+1 g(x)=1/x we need to find foG (f of g) [composite functions].
  50. lep11

    Expressing defined integral as composition of differentiable functions

    Homework Statement Let ##f(t)=\int_{t}^{t^2} \frac{1}{s+\sin{s}}ds,t>1.##Express ##f## as a composition of two differentiable functions ##g:ℝ→ℝ^2## and ##h:ℝ^2→ℝ##. In addition, find the derivative of ##f## (using the composition). Homework EquationsThe Attempt at a Solution Honestly, I have...
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