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. Essence

    Java Defining variables in the context of implicit functions in java

    Sorry for the disturbance, So I have been looking (without success) for a way to define a variable within an implicit function in Java. What I really mean by this is I have the equation: In this function my program will give me all of the values except for px. I have tried rearranging the...
  2. C

    Do wave functions go to zero at ~ct?

    Suppose you have a free election and you make a measurement of its position r_0 at time t = 0. You then wait some time t required for the wave function to evolve out of its collapsed eigenstate. The electron now supposedly has a wave function expanding to infinity in all directions, albeit with...
  3. M

    Motivation of sin and cos functions

    Is there a way to motivate the sinus and cosinus functions by looking at their Taylor expansion? Or equivalently, is there a way to see that complex numbers adds their angles when multiplied without knowledge of sin and cos?
  4. N

    Proof using hyperbolic trig functions and complex variables

    1. Given, x + yi = tan^-1 ((exp(a + bi)). Prove that tan(2x) = -cos(b) / sinh(a)Homework Equations I have derived. tan(x + yi) = i*tan(x)*tanh(y) / 1 - i*tan(x)*tanh(y) tan(2x) = 2tanx / 1 - tan^2 (x) Exp(a+bi) = exp(a) *(cos(b) + i*sin(b))[/B]3. My attempt: By...
  5. D

    Sum of Two Periodic Orthogonal Functions

    Homework Statement This problem is not from a textbook, it is something I have been thinking about after watching some lectures on Fourier series, the Fourier transform, and the Laplace transform. Suppose you have a real valued periodic function f with fundamental period R and a real valued...
  6. J

    Are functions partly defined by their domains and codomains?

    I just finished working through compositions of functions, and what properties the inner and outer functions need to have in order for the whole composition to be injective or surjective. I checked Wikipedia just to make sure I'm right in thinking that for a composition to be injective or...
  7. S

    Irrational Roots Theorems for Polynomial Functions

    Is any Irrational Roots Theorem been developed for polynomial functions in the same way as Rational Roots Theorems for polynomial functions? We can choose several possible RATIONAL roots to test when we have polynomial functions; but if there are suspected IRRATIONAL roots, can they be found...
  8. W

    Integrals of the Bessel functions of the first kind

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  9. O

    MHB Decreasing nonnegative sequence and nonincreasing functions

    Let $\{p_n\}$ be a nonnegative nonincreasing sequence and converges to $p \ge 0$. Let $f : [0,\infty)\to[0,\infty)$ be a nondecreasing function. So, since f is a nondecreasing function, $f(p_n)>f(p)>0$. How did this happen?
  10. P

    Inverse trigonometric functions

    I am familiar with the importance of the following inverse circular/hyperbolic functions: ##\sin^{-1}##, ##\cos^{-1}##, ##\tan^{-1}##, ##\sinh^{-1}##, ##\cosh^{-1}##, ##\tanh^{-1}##. However, I don't really get the point of ##\csc^{-1}##, ##\coth^{-1}##, and so on. Given any equation of the form...
  11. S

    A difficult problem on functions

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  12. P

    Inverse hyperbolic functions (logarithmic form)

    To express the ##\cosh^{-1}## function as a logarithm, we start by defining the variables ##x## and ##y## as follows: $$y = \cosh^{-1}{x}$$ $$x = \cosh{y}$$ Where ##y ∈ [0, \infty)## and ##x ∈ [1, \infty)##. Using the definition of the hyperbolic cosine function, rearranging, and multiplying...
  13. B

    How to correctly convolve two delta functions?

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  14. A

    Transforming Functions: Solving g(x) = 2f(-x+(3/2))

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  15. RyanH42

    Geometry Vector Functions Searching Video Lecture

    Hi I am studying Differantial Geometry.My textbook is Lipschultz Differantia Geometry and I am in chapter two.I undersand the basic idea of Vector function but I wanI to gain more information.Is there any video lectures about vector functions.I found this...
  16. D

    Arguments of exponential and trig functions

    What can be said about the arguments of the exponential functions and trig functions ? Can the argument be a vector or must it be a scalar ? If it can only be a scalar must it be dimensionless ?
  17. S

    Optimization methods with bivariate functions

    Hi, I have the following equation: f(z)=g(z)+b*u(z) where z=(x,y) i.e. bivariate,b is a parameter, u(z) the uniform distribution and g(z) a function that represents distance. By considering for a momment b=0, min(f(z)) can give me the location of the minimum distance. However because I want...
  18. Jaco Viljoen

    Let f, g and h be functions defined below:

    Homework Statement f(x)=(√x^2-3x+2)/(2x-3), g(x)=3/(√(x+3)) and h(x)=(x^2-5x+6)/(x-2) which of the following are true: A)Df={x∈ℝ:x≤1 or x≥2} B)Dg={x∈ℝ:x≥-3} C)Dh=ℝ Homework EquationsThe Attempt at a Solution I am using substitution here by replacing the x by the parameters specified and...
  19. Jaco Viljoen

    Let f, g and h be functions defined as follows:

    Homework Statement f(x)=(√x2-3x+2)/(2x-3), g(x)=3/(√x+3) and h(x)=(x2-5x+6)/(x-2) which of the following are true: A)Df={x∈ℝ:x≤1 or x≥2} B)Dg={x∈ℝ:x≥-3} C)Dh=ℝ Homework EquationsThe Attempt at a Solution I am only attempting now,
  20. S

    Modeling Functions.... real life business, money, probability, etc.

    Ok. I'm now studying economics and applied math, and I'm currently wanting to know what book or online resource could help me in learning how to model real life situations into functions. Most math and econ textbooks are garbage at this. I'll be more specific. In my study of Microeconomics...
  21. rolotomassi

    Green's Functions and Feynman Diagrams

    I've been learning about Greens functions. I'm familiar with how to find them for different differential operators and situations but far from fully understanding them. We were shown in lecture how they can be used to obtain a perturbation series, leading to Feynman diagrams which represent...
  22. Rafael de Gomes

    Maple - Finding maximum of maximums in different functions

    Hello, I've been trying to come up with a short way of writing the code. What I'm trying to do is: I have 11 equations, each of which have a defined minimum and maximum. I'm trying to find the highest maximum out of all of them and I need to know which one it is. The highest as in farthest...
  23. T

    MHB Solve Multiline Function w/2 Variables - Explanation Here

    Can anyone explain how to solve a multiline function with two variables? Please see attached.
  24. V

    Find Intersections of Trig Functions with different periods

    There are 2 trig functions on the same set of axis. f(x)=600sin(2π3(x−0.25))+1000 and f(x)=600sin(2π7(x))+500 How do I go about finding the points of intersections of the two graphs? This was from a test I had recently and didn't do too well on,so any help would be much appreciated. I started...
  25. evinda

    MHB Distances between the functions

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  26. S

    Square-integrable functions as a vector space

    Homework Statement (a) Show that the set of all square-integrable functions is a vector space. Is the set of all normalised functions a vector space? (b) Show that the integral ##\int^{a}_{b} f(x)^{*} g(x) dx## satisfies the conditions for an inner product. Homework Equations The main...
  27. K

    Where does band index come from in block wave functions?

    Can you show me as explicitly as possible through equations?
  28. Kyuutoryuu

    Are hyperbolic functions used in Calculus 3?

    More than just a few problems that happen to pop up in the textbook, I mean.
  29. K

    Finding poles of complex functions

    I am trying to calculate a pole of f(z)=http://www4b.wolframalpha.com/Calculate/MSP/MSP86721gicihdh283d613000033ch4ae4eh37cbd4?MSPStoreType=image/gif&s=35&w=44.&h=40. . The answer in the textbook is: Simple pole at...
  30. Chrono G. Xay

    'Wheel-like' Mathematics (Modulating Trig Functions?)

    As part of a personal musicology project I found myself with the mathematical model of a geometry which utilizes the equation a*(a/b)sin(pi*x) The only problem with this is that I need to take the integral from -1/2 <= x <= 1/2, and according to Wolfram Alpha no such integral exists. I can...
  31. A

    Why do collapse functions change form in master equations?

    In the GRW spontaneous collapse model (for example) the wave-function evolves by linear Schrödinger equation, except, at random times, wave-function experiences a jump of the form: \psi_t(x_1, x_2, ..., x_n) \rightarrow \frac{L_n(x)\psi_t(x_1, x_2, ..., x_n)}{||\psi_t(x_1, x_2, ..., x_n)||}...
  32. K

    Why does an inverse exist only for surjective functions?

    In other words, why does a function have to be onto (or surjective, i.e. Range=Codomain) for its inverse to exist?
  33. M

    MHB Can We Enumerate All Primitive Recursive Functions?

    Hey! :o Can we enumerate the primitive recursive functions?
  34. S

    Orthogonality relations for Hankel functions

    Where can I find and how can I derive the orthogonality relations for Hankel's functions defined as follows: H^{(1)}_{m}(z) \equiv J_{n}(z) +i Y_{n}(z) H^{(2)}_{m}(z) \equiv J_{n}(z) - i Y_{n}(z) Any help is greatly appreciated. Thanks
  35. gonadas91

    Can We Determine a Complex Function from Its Poles Alone?

    Suppose we have a complex function f(z) with simple poles on the complex plane, and we know exactly where these poles are located (but we don't know how the function depends on z) Is there any way to build up the exact form of f(z) just from its poles?
  36. H

    Raabe's test says Legendre functions always converge?

    The Legendre functions are the solutions to the Legendre differential equation. They are given as a power series by the recursive formula (link [1] given below): ##\begin{align}y(x)=\sum_{n=0}^\infty a_n x^n\end{align}## ##\begin{align}a_{n+2}=-\frac{(l+n+1)(l-n)}{(n+1)(n+2)}a_n\end{align}##...
  37. ChrisVer

    Positive integral for two functions

    I am feeling stupid today...so: Is the following statement true? \int_{-\infty}^{+\infty} f(x) g(x) \ge 0 if f(x) \ge 0 and 0< \int_{-\infty}^{+\infty} g(x) < \infty (integral converges)? If yes, then how could I show that? (It's not a homework) I am trying to understand how the...
  38. gonadas91

    Meromorphic functions expansions

    Hi, is there any good book or table with all the known expansions for meromorphic functions in the complex plane¿ (Using Mittag-Leffer theorem to express the function as a sum of its poles) I am trying to evaluate an infinite sum which seems rather complicated and I wonder if there is something...
  39. P

    Delay complexity of Boolean functions

    Homework Statement I don't really understand how/why every Boolean function of n variables may be implemented with a delay complexity of O(n). Could someone please try and explain? Homework EquationsThe Attempt at a Solution I was trying to show it using minterms (SOP). There is a maximum of...
  40. A

    Greens functions from path integral

    Let me post this question again in a slightly modified form. On the attached picture the path integral for the partion function: Z = Tr(exp(-βH)) Now according to what it says on the picture it should be easy from this to get the Green's function in the path integral formalism. The Green's...
  41. DavitosanX

    What's a good book for Green's functions?

    Hello! I'm currently taking a course on partial differential equations, and we're using Asmar's textbook. We've reached Green's functions, and even though Asmar is a great book, I feel like I need a deeper study of the subject. Which book would you recommend to help me better grasp the theory...
  42. C

    Rational functions: combine and simplify terms

    Homework Statement (4a/a+4)+(a+2/2a) Homework Equations Just combine and then factor out The Attempt at a Solution It's actually fairly simple, but I'm having difficulty at the end. /multiply each term by opposite denominator 4a(2a)/a+4(2a) + a+2(a+4)/2a(a+4) /combine 4a(2a)+(a+2)(a+4) /...
  43. T

    Trig functions in terms of x,y, and r?

    I work a good deal better when the equation is in x and y form, is it possible to set up a trig expression like 5Cos(x)/(Sin(x)-1)and substitute the proper x or y equivalent so long as I remember to replace the trig identities later when the problem is finished? Or can you just not solve these...
  44. J

    MATLAB Vector-valued functions in MATLAB

    I'm trying to create a vector field plot of an equation in x and y. Basically, I would like to create a function F(x, y) = p(x, y)i + q(x, y)j that defines a force field, and have the field direction and magnitude plotted at points in the x-y plane, and both components of the vector are...
  45. K

    Jacobi elliptic functions with complex variables

    I am trying to solve a Duffing's equation ##\ddot{x}(t)+\alpha x(t)+\beta x^3(t)=0## where ##\alpha## is a complex number with ##Re \alpha<0## and ##\beta>0##. The solution can be written as Jacobi elliptic function ##cn(\omega t,k)##. Then both ##\omega## and ##k## are complex. The solution to...
  46. gonadas91

    Change of sign on Green's functions (Maths problem)

    Hi, I am trying to solve a model where Non-interacting Green functions take part it. It has happened something that is spinning my head and I hope someone could help. The non interacting Green function for a chanel of electrons is...
  47. M

    Finding Points of Intersection for Two Functions

    Homework Statement The problem ask for points of intersection of two functions Homework Equations 1: 2x+y-4=0 2: (y^2)-4x=0 The Attempt at a Solution My attempt of solution its in a picture attached below... I get stuck in this two equations 1: ((y^2)/4)+(y/2)-2=0 2: square...
  48. M

    MHB Recursive and Primitive recursive functions

    Hey! :o According to the book that I'm reading, we can define the $\mu-$recursive functions inductively, as follows: The constant, projection, and successor functions are all $\mu-$recursive. If $g_1, \dots , g_m$ are $n-$variable $\mu-$recursive functions and $h$ is an $m-$variable...
  49. W

    Trigonometric functions (identity&equations)

    1) Problem: given that x is an obtuse angle for which cos^2x/(1 + 5sin^2x) = 8/35, find the value of cosx/(1 - 5 sin x) without evaluating x. 2) Relevent equations: sin(-x) = - sin x cos(-x) = cos x sin(180° - x) = sin x cos(180° - x) = - cos x sin^2x + cos^2x = 1 3) Attempt: cos^2x/(1 +...
  50. B

    Physical insight into integrating a product of two functions

    I was wondering what the physical insight is of integrating a product of two functions. When we do that for a Fourier transform, we decompose a function into its constituent frequencies, and that's because the exponential with an imaginary x in the transform can be seen as a weighting function...
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