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. Math Amateur

    I Composition of Two Continuous Functions .... Browder, Proposition 3.12

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ... I need some help in understanding the proof of Proposition 3.12...
  2. Math Amateur

    MHB Composition of Two Continuous Functions .... Browder, Proposition 3.12 .... ....

    I am reading Andrew Browder's book: "Mathematical Analysis: An Introduction" ... ... I am currently reading Chapter 3: Continuous Functions on Intervals and am currently focused on Section 3.1 Limits and Continuity ... ... I need some help in understanding the proof of Proposition 3.12...
  3. F

    Sequence of integrable functions (f_n) conv. to f

    ##\textbf{Attempt at solution}##: If I can show that ##f## is integrable on ##[a,b]##, then for the second part I get : Let ##\frac{\varepsilon}{b-a} > 0##. By definition of uniform convergence, there exists ##N = N(\varepsilon) > 0## such that for all ##x \in [a,b]## we have ##\vert f(x) -...
  4. W

    A Conditions to extend functions Continuously into the Boundary (D^1/S^1)

    Other than for null-homotopic maps, which continuous maps defined on ##D^1 \rightarrow D^1## (Open disk)extend continuously to maps ##B^1 \rightarrow B^1 ## ,(##B^1## the closed disk) which maps can be extended in opposite direction, i.e., continuous maps ## f: S^1 \rightarrow S^1 ## that...
  5. S

    I Which class of functions does 1/x belong to?

    For historical reasons the hyperbola always was considered to be one of the «classical» curves. The function, obviously, does not belong to C0. Apparently, is does not fit L2 or any other Lp? What is the smallest class?
  6. Math Amateur

    MHB Limits of Functions ....Conway, Proposition 2.1.2 .... ....

    I am reading John B. Conway's book: A First Course in Analysis and am focused on Chapter 2: Differentiation ... and in particular I am focused on Section 2.1: Limits ... I need help with an aspect of the proof of Proposition 2.1.2 ...Proposition 2.1.2 and its proof read as follows: In...
  7. Math Amateur

    I Limits of Functions ....Conway, Proposition 2.1.2 .... ....

    I am reading John B. Conway's book: A First Course in Analysis and am focused on Chapter 2: Differentiation ... and in particular I am focused on Section 2.1: Limits ... I need help with an aspect of the proof of Proposition 2.1.2 ...Proposition 2.1.2 and its proof read as follows: In the...
  8. karush

    MHB 4.1.310 AP calculus Exam Area under to functions

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  9. davidfur

    How to use functions defined in an external header-only library?

    Hey guys, I have written a C++ code which is based on two main classes: Particle and Group. Each Group contains a set of Particle(s), each Particle is defined by a set of coordinates, and has an associated energy and force (the energy/force evaluation is done by calling an external program). I...
  10. tworitdash

    A Integral of 2 Bessel functions of different orders

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  11. S

    I Nested expressions as compositions of functions?

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  12. Srinath

    Timoshenko Beam Theory (Violin String Shape Functions)

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  13. K

    I The sum of these functions equals a constant

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  14. Math Amateur

    MHB Help Understanding Andrew Browder's Proposition 8.14

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  15. Math Amateur

    MHB Understand Andrew Browder's Prop 8.13: Math Analysis Introduction

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  16. tworitdash

    A Integration of Bessel's functions

    I can only find a solution to \int_{0}^{r} \rho J_m(a\rho) J_n(b\rho) d\rho with the Lommel's integral . The closed form solution to \int_{0}^{r}\frac{1}{\rho} J_m(a\rho) J_n(b\rho) d\rho I am not able to find anywhere. Is there any way in which I can approach this problem from scratch...
  17. D

    Voltage and current as functions of time for a series RL circuit

    I already found ##I(t)## using Kirchhoff's laws, I got the equation ##V-RI-L\frac{dI}{dt}=0\Rightarrow L\frac{dI}{dt}=V-RI## then I solved the differential equation getting ##I(t)=\frac{V}{R}\left[1-e^{-\frac{R}{L}t}\right]##. My problem is founding the voltage as a function of time ##V(t)##, I...
  18. N

    MHB Decide volume given two functions

    Sorry if i made any language errors, english is not my first language. Question: An area in the first quadrant (x=>0,y=>0) is limited by the axis and the graphs to the functions f(x)=x^2-2 and g(x)=2+x^2/4. When the area rotates around the y-axis a solid is created. Calculate the volume of...
  19. S

    Mathematica Plotting an array of functions in different colors

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  20. V

    MHB Help with Functions - Linearization

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  21. M

    A Finding eigenvalues with spectral technique: basis functions fail

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  22. L

    Green's functions (Fourier Series)

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  23. D

    I Odd/even functions and fractional indices

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  24. S

    MHB Multiple Transformations of Functions

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  25. N

    ProProgram for drawing functions

    Hello, good morning. I would like to know if someone knows a program to be able to draw the functions in the same way as the one shown in the image and also allow me to point out an enclosure formed by them without having to use inequalities to do so. Thank you very much for everything beforehand.
  26. Math Amateur

    MHB Diiferentiability of Functions of a Complex Variable .... Markushevich, Theorem 7.1 .... ....

    I am reading the book: "Theory of Functions of a Complex Variable" by A. I. Markushevich (Part 1) ... I need some help with an aspect of the proof of Theorem 7.1 ...The statement of Theorem 7.1 reads as follows: At the start of the above proof by Markushevich we read the following: "If f(z)...
  27. R

    Exploring Trigonometric Functions & Physics: Velocity, Distance, & Dimension

    Hello, It has been a long time since I first looked at this, so thought I might ask for some help in clarifying this problem: Is an equation of the form --> Velocity = (Distance) * (Trigonometric function) a valid one in physics? If so, what is the relationship of trigonometric functions...
  28. S

    I Rational functions in one indeterminate - useful concept?

    The examples of "formal" power series and polynomials in one indeterminate are familiar and useful in algebra. However, I don't recall the example of rational functions (ratios of polynomials) in one indeterminate being used for anything. Is that concept useful? - or trivial? -or equivalent...
  29. Haorong Wu

    Should spacial functions be involved when calculating <Sx>?

    I have two different solutions, and I do not know which one is correct and why the other one is wrong. Solution 1. In the ##L_z## space, the spin state is ##\begin{pmatrix} \sqrt { \frac 4 5} \\ \sqrt { \frac 1 5} \end{pmatrix}##, and ##S_x=\frac \hbar 2 \begin{pmatrix} 0& 1 \\ 1& 0...
  30. P

    B Correct way to write multiple argument functions

    Hi, This is on the wikipedia entry for the Euler Lagrange equation. Here is a link. https://en.wikipedia.org/wiki/Calculus_of_variations#Euler%E2%80%93Lagrange_equation The notation I am confused about is this: Aren't the y(x) and the y'(x) unnecessary to list as arguments when x is...
  31. I

    Family of functions satisfying a particular criterion

    Since, both the domain and the range is set of integers, we must have just operations of addition and multiplication only in the function. That means, function should be some kind of a polynomial. Plugging ##a=0## and ##b=0##, I can deduce that ##3 f(0) = f(f(0))##. Also I can deduce that ##3...
  32. R

    Drawing Graphs: Concentric Circles & Straight Lines

    Hi, how would I go about drawing these two graphs? and The first one would be concentric circles with the centre at (0,0). The second one would be straight lines through (0,0). Is this correct? Also, what happens at ln(0) = constant for the first graph and x = 0 for the second graph...
  33. Math Amateur

    MHB Verify Gamelin's Remark: Complex Square and Square Root Functions

    I am reading Theodore W. Gamelin's book: "Complex Analysis" ... I am focused on Chapter 1: The Complex Plane and Elementary Functions ... I am currently reading Chapter 1, Section 4: The Square and Square Root Functions ... and need some help in verifying a remark by Gamelin ... ... The...
  34. snatchingthepi

    Max and min functions in spherical expansions

    I'm trying to solve the vector potential of a solid rotating sphere with a constant charge density. I'm at a point where I'm performing the final integral that looks like $$ -\left( \frac {\mu_0 i} {3} \right) \sqrt{\frac 3 {2\pi}} \frac {q\omega}{R^3} Y_{1,1} \int_0^R (r')^3 \frac {r_<}...
  35. M

    Potential functions for separation and isochronic gauges

    Most potentials in physics are expressed as a radius or another geometric norm/gauge. I am looking to understand the significance of the choice of potential functions for force/pressure separation in harmonic analysis before this creates a topology. To my understanding this is the decision of...
  36. M

    Vector space - polynomials vs. functions

    As per source # 1 ( link below), when treating polynomials as vectors, we use their coefficients as vector elements, similar to what we do when we create matrices to represent simultaneous equations. However, what I noticed in Source #2 was that, when functions are represented as vectors, the...
  37. Adgorn

    Limit of the remainder of Taylor polynomial of composite functions

    Since $$\lim_{x \rightarrow 0} \frac {R_{n,0,f}(x)} {x^n}=0,$$ ##P_{n,0,g}(x)## contains only terms of degree ##\geq 1## and ##R_{n,0,g}## approaches ##0## as quickly as ##x^n##, I can most likely prove this using ##\epsilon - \delta## arguments, but that seems overly complicated. I also can't...
  38. Danny Boy

    A Correlation functions of quantum Ising model

    I have a single technical question regarding a statement on page 7 of the paper "Dynamical quantum correlations of Ising models on an arbitrary lattice and their resilience to decoherence". The paper up until page 7 defines a general correlation function ##\mathcal{G}## of a basic quantum Ising...
  39. jk22

    I How to integrate with functions of differentials?

    I fell upon such a wrting : $$du=tan(d\theta)$$ How to integrate this ? I didn't try numerically but I thought of expanding the tangeant in series but then should for example $$\int d\theta^2$$ be understood as a double integration ?
  40. karush

    MHB Is Tikx the Solution for Creating Inverse Functions in Overleaf?

    ok I have been trying to cut and paste in packages and code to get a simple inverse function to plot but nutin shows up and get error message. if possible I would like no grid but an xy axis with tick only where the graph goes thru the axis and of course a dashed line of x=y some of the...
  41. karush

    MHB Simplifying $\cot^2(x)-\csc^2(x)$: 1

    Write $\cot^2(x)-\csc^2(x)$ In terms of sine and cosine and simplify So then $\dfrac{\cos ^2(x)}{\sin^2(x)} -\dfrac{1}{\sin^2(x)} =\dfrac{\cos^2(x)-1}{\sin^2(x)} =\dfrac{\sin^2(x)}{\sin^2(x)}=1$ Really this shrank to 1 Ok did these on cell so...
  42. karush

    MHB 242 Derivatives of Logarithmic Functions of y=xlnx-x

    $\tiny{from\, steward\, v8\, 6.4.2}$ find y' $\quad y= x\ln{x}-x$ so $\quad y'=(x\ln{x})'-(x)'$ product rule $\quad (x\ln{x})'=x\cdot\dfrac{1}{x}+\ln{x}\cdot(1)=1+\ln{x}$ and $\quad (-x)'=-1$ finally $\quad \ln{x}+1-1=\ln{x}$...
  43. dRic2

    I Bloch functions and momentum of electrons in a lattice

    Hi, I'm a bit confused about Bloch functions. This is what, I think, I understood: can someone please tell me what's wrong? From Bloch's theorem we know that the wave-function of an electron inside a periodical lattice can be written as ##ψ_k(r)=u_k(r)e^{ik⋅r}##. We hope that far from a lattice...
  44. J

    A Creation/annihilation operators and trigonometric functions

    Hello everyone, I have noticed a striking similarity between expressions for creation/annihilation operators in terms position and momentum operators and trigonometric expressions in terms of exponentials. In the treatment by T. Lancaster and S. Blundell, "Quantum Field Theory for the Gifted...
  45. Math Amateur

    MHB Exploring Theorem 4.29: Compact Metric Spaces & Inverse Functions

    I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ... I am focused on Chapter 4: Limits and Continuity ... ... I need help in order to fully understand the example given after Theorem 4.29 ... ... Theorem 4.29 (including its proof) and the following example read as...
  46. Math Amateur

    MHB Functions Continuous on Comapct Sets .... Apostol, Theorem 4.25 ....

    I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ... I am focused on Chapter 4: Limits and Continuity ... ... I need help in order to fully understand the proof of Theorem 4.25 ... ... Theorem 4.25 (including its proof) reads as follows: In the above proof by...
  47. Math Amateur

    MHB Limits of Complex Functions .... Zill & Shanahan, Theorem 3.1.1/ A1

    I am reading the book: Complex Analysis: A First Course with Applications (Third Edition) by Dennis G. Zill and Patrick D. Shanahan ... I need some help with an aspect of the proof of Theorem 3.1.1 (also named Theorem A1 and proved in Appendix 1) ... The statement of Theorem 3.1.1 (A1) reads...
  48. P

    MHB Inverse Functions: Reflection of f(x) & g(x) Logic

    Can anyone explain the logic behind the answer? Taken from HiSet free practice test
  49. O

    I Higher-Order Time Correlation Functions of White Noise?

    Suppose I have Gaussian white noise, with the usual dirac-delta autocorrelation function, <F1(t1)F2(t2)> = s2*d(t1-t2)*D12 Where s is the standard deviation of the Gaussian, little d is the delta function, and big D is the kronecker delta. For concreteness and to keep track of units, say F...
  50. F

    A The partial derivative of a function that includes step functions

    I have this function, and I want to take the derivative. It includes a unit step function where the input changes with time. I am having a hard time taking the derivative because the derivative of the unit step is infinity. Can anyone help me? ##S(t) = \sum_{j=1}^N I(R_j(t)) a_j\\ I(R_j) =...
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