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. A

    MHB Calculating g(x) for Y=f(x) Passing Through Points

    Y=f(x) which passes through points: (-1,3) and (0,2) and (1,0) and (2,1) and (3,5) second function is defined: g(x)=2f(x-1) Calculate g(0) Calculate g(1) Calculate g(2) Calculate g(3)
  2. YoungPhysicist

    Big integer arithmetic functions

    NOTE:This is not a homework question! This is just a topic that I like very much,but don’t have the programming ability to do many of them.That’s why I post this thread. C++ is a language without built-in big integer calculation functions,so building ones that can do such job is a great way to...
  3. S

    Does anybody know of any good books for math? (functions)

    Does anybody know any good books to learn math?
  4. D

    How to use the window functions on a signal in MATLAB?

    Homework Statement I am suppose to write a program that compares the FFT (Fast Fourier Transform Diagrams) of a sampled signal without the use of a window function and with it. The window function should be as long as the signal and the signal should have N points, N chosen as to not cause...
  5. E

    Proving closure of square integrable functions.

    I'm trying to prove that the set of all square integrable functions f(x) for which ∫ab |f(x)|^2 dx is finite is a vector space. Everything but the proof of closure is trivial. To prove closure, obviously we should expand out |f(x)+g(x)|^2, which turns our integral into one of |f(x)|^2 (finite)...
  6. Math Amateur

    MHB Graphs of Functions Covering Unit Circle .... Hubbard & Hubbard, Example 3.1.5 .... ....

    I am reading the book: "Vector Calculus, Linear Algebra and Differential Forms" (Fourth Edition) by John H Hubbard and Barbara Burke Hubbard. I am currently focused on Section 3.1: Manifolds ... I need some help in order to understand Example 3.1.3 ... ... Example 3.1.3 reads as follows:In...
  7. YoungPhysicist

    B A rookie question for integrals of polynomial functions

    $$\int x^2+3 = \frac{x^3}{3}+3x+C$$ I can get the front two part by power rule, but what is the C doing there? Wolframalpha suggested it should be a constant, but what value should it be? Sorry for asking rookie questions:-p
  8. R

    MHB Trig Functions: When Plugging in x Returns x

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  9. M

    MHB Convex Functions: Info on Minima, Reconstructing Original Functions

    If you would allow me to ask... if i have two convex functions , and i was to place one inside the other, i.e. convolute them...what could be said in general about the resultant function. what information about the original functions can be taken from the positions of the minima. and is there...
  10. Robin04

    I Understanding the definition of continuous functions

    Definition: A function f mapping from the topological space X to the topological space Y is continuous if the inverse image of every open set in Y is an open set in X. The book I'm reading (Charles Nash: Topology and Geometry for Physicists) emphasizes that inversing this definition would not...
  11. C

    Factoring Combinatorial Functions

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  12. J

    MHB Increase and decrease functions

    Hello I have tried to resolve an exercise which is asking how the graph is modified according to the variables into the function. I would appreciate any help since accordin to my udnerstanding the function should increase Please, follow below: Suppose y0 is the y-coordinate of the point of...
  13. M

    I Fitting functions based on imperfect data

    I have a set of values and I'm trying to come up with functions to fit that data. Here is what I know about the data: It is rounded down / floored to the nearest significant digit (i.e. 1 for v1 and v3, 0.1 for v2). Columns v1 and v3 look linear (e.g. first order polynomial). Column v2 looks...
  14. PKSharma

    I Can we have a pasting lemma for uniform continuous functions

    In analysis, the pasting or gluing lemma, is an important result which says that two continuous functions can be "glued together" to create another continuous function. The lemma is implicit in the use of piecewise functions. Can we have a similar situation for uniform continuous functions?
  15. S

    MHB Functions and Relations: Proving R is a Function from A to B

    Let R\subseteq A*B be a binary relation from A to B , show that R is a function if and only if R^-1(not) R \subseteq idB and Rnot aR^-1 \supseteq both hold. Remember that Ida(idB) denotes the identity relation/ Function {(a.a)|a€ A} over A ( respectively ,B) Please see the attachment ,I...
  16. ranga519

    MHB What Comparison Sign To Assert f^(-1)(f(A))? A True?

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  17. Math Amateur

    I The Chain Rule for Multivariable Vector-Valued Functions ....

    I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ... I am focused on Chapter 12: Multivariable Differential Calculus ... and in particular on Section 12.9: The Chain Rule ... ... I need help in order to fully understand Theorem 12.7, Section 12.9 ... Theorem 12.7...
  18. Math Amateur

    MHB The Chain Rule for Multivariable Vector-Valued Functions .... ....

    I am reading Tom M Apostol's book "Mathematical Analysis" (Second Edition) ...I am focused on Chapter 12: Multivariable Differential Calculus ... and in particular on Section 12.9: The Chain Rule ... ...I need help in order to fully understand Theorem 12.7, Section 12.9 ...Theorem 12.7...
  19. T

    Is this question incomplete? Regarding entire functions....

    Homework Statement Let ##F## be an entire function such that ##\exists## positve constants ##c## and ##d## where ##\vert f(z)\vert \leq c+d\vert z\vert^n, \forall z\in \Bbb{C}##. Is this question incomplete? My complex analysis course is not rigorous at all and this came up on a past final...
  20. Math Amateur

    MHB Differentiability of Multivariable Vector-Valued Functions .... ....

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  21. K

    Volume of revolution, region bounded by two functions

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  22. karush

    MHB Limit $\frac{f(x)}{g(x)}$: Solve w/ L'H Rule

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  23. M

    MHB Intersection points of two quadratic functions

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  24. J

    MHB Properties of Functions of Bounded Variation

    Sorry for all the questions. Reviewing for my midterm next week. Fun fun. If someone could take a look at my proof for (a) and help me out with (b) that'd be awesome! (a) Let $\Delta$ be a partition of $[a, b]$ that is a refinement of partition $\Delta'$. For a real-value function $f$ on $[a...
  25. A

    Uniform convergence of a sequence of functions

    Homework Statement This is a translation so sorry in advance if there are funky words in here[/B] f: ℝ→ℝ a function 2 time differentiable on ℝ. The second derivative f'' is bounded on ℝ. Show that the sequence on functions $$ n[f(x + 1/n) - f(x)] $$ converges uniformly on f'(x) on ℝ...
  26. J

    MHB Bounded Variation - Difference of Functions

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  27. F

    Showing Uniform Convergence of Cauchy Sequence of Functions

    Homework Statement Let ##X \subset \mathbb{C}##, and let ##f_n : X \rightarrow \mathbb{C}## be a sequence of functions. Show if ##f_n## is uniformly Cauchy, then ##f_n## converges uniformly to some ##f: X \rightarrow \mathbb{C}##. Homework Equations Uniform convergence: for all ##\varepsilon >...
  28. M

    A Green's Function Approach for ODE with Boundary Conditions - Why the Difference?

    Hi PF! The ODE $$g''(x) + (1-k^2)g(x) = f(x)\\ g(0) = y(\pi/3) = 0$$ where ##f(x)## is a forcing function and ##k \in \mathbb N## is a constant has a Green's function via variation of parameters as $$ G_L = \frac{L(y)R(x)}{W} : 0<x<y<\pi/3\\ G_R = \frac{L(x)R(y)}{W} : 0<y<x<\pi/3 $$ with...
  29. R

    I Differential of the coordinate functions

    Hello folks, I'm glad that I discovered this forum. :) You might save me. I'm hearing right now differential geometry and am having some problems with the subject. May you explain me the follwoing. We had the special case of the i-th projection. My lecturer now posited that the differential of...
  30. M

    A Variation of parameters, Green's functions, Wronskian

    Hi PF! I am trying to solve an ODE by casting it as an operator problem, say ##K[y(x)] = \lambda M[y(x)]##, where ##y## is a trial function, ##x## is the independent variable, ##\lambda## is the eigenvalue, and ##K,M## are linear differential operators. For this particular problem, it's easier...
  31. J

    MHB Derivatives of trigonometric functions

    The answer for derivative of y=5tanx+4cotx is y'=-5cscx^2. But how come on math help the answer is 5sec^2x-4csc^2x? I have a calculus test coming up and I really would appreciate if someone could explain! - - - Updated - - - Oh nvm I see my mistake!
  32. ubergewehr273

    Question about a function of sets

    Let a function ##f:X \to X## be defined. Let A and B be sets such that ##A \subseteq X## and ##B \subseteq X##. Then which of the following are correct ? a) ##f(A \cup B) = f(A) \cup f(B)## b) ##f(A \cap B) = f(A) \cap f(B)## c) ##f^{-1}(A \cup B) = f^{-1}(A) \cup f^{-1}(B)## d) ##f^{-1}(A \cap...
  33. J

    What is the domain of f(f(x)) for f(x)= x/(1+x)?

    Homework Statement f(x)= x/(1+x) What is f(f(x)) and what is its domain. 2. The attempt at a solution I found f(f(x))= x/(1+2x) and the domain: (-∞,-1/2)∪(-1/2,∞) , but it is saying that I have the wrong domain. What mistake have I made? My process for finding domain: 1. Find the domain...
  34. opus

    B Limits on Composite Functions- Appears DNE but has a limit

    Please see my attached image, which is a screenshot from Khan Academy on the limits of composite functions. I just want to check if I'm understanding this correctly, particularly for #1, which has work shown on the picture. Now my question: We are taking the limit of a composition of...
  35. J

    MHB Proving of continuous functions

    Appreciate the help needed for the attached question. Thanks!
  36. B

    A Residues of gamma functions

    Hi members, I have a problem with the computation of residues involve the gamma functions. (see attached Pdf file) Can you show me for the first residue with the arrows, or give a hint or a link. Thank you
  37. Philip Robotic

    Integration that leads to logarithm functions problem

    Hi everyone, So I am a high school student and I am learning calculus by myself right now (pretty new to that stuff still). Currently I am working through some problems where integration leads to logarithm functions. While doing one of the exercises I noticed one thing I don't understand. I...
  38. F

    Hi, I have a quick question about inverse functions.

    One of our homework problem asks: If f is a one-to-one function such that f(-3)=5 , find x given that f^-1 (5)=3x-1. Here's how I attempted to solve the problem: -3=3x-1 3x=-2 x=-2/3 Is this the correct way to solve the problem?
  39. A

    I Green's functions in QFT for the gifted amateur

    Hello, I am reading the book QFT for the gifted amateur and I have a question concerning how to go from the wave function picture to the Green's function as defined by equations (16.13) and (16.18) at page 147. ## \phi(x,t_{x}) = \int dy G^{+}(x,t_{x},y,t_{y})\phi(y,t_{y}) ##...
  40. U

    Is f Injective? Understanding the Composition of Functions

    Homework Statement Let A, B, C be finite sets such that A and B have the same number of elements, that is, |A| = |B|. Let f : A → B and g : B → C be functions. (a) Suppose f is one-to-one. Show that f is onto. (b) Suppose g ◦ f is one-to-one. Show that g is one-to-one.Homework EquationsThe...
  41. bhobba

    Non Computable Functions And Godel's Theorem

    Hi All I normally post on the QM forum but also have done quite a bit of programming and did study computer science at uni. I have been reading a book about Ramanujan and interestingly he was also good friends with Bertrand Russell. You normally associate Russell with philosophy but in fact...
  42. Pushoam

    Functions forming a vector space

    Homework Statement 1.1.3 1) Do functions that vanish at the endpoints x=0 and L=0 form a vector space? 2) How about periodic functions? obeying f(0)=f(L) ? 3) How about functions that obey f(0)=4 ? If the functions do not qualify, list what go wrong.Homework Equations The Attempt at a...
  43. C

    Fourier Analysis and the Significance of Odd and Even Functions

    Homework Statement Q1. a) In relation to Fourier analysis state the meaning and significance of 4 i) odd and even functions ii) half-wave symmetry {i.e. f(t+π)= −f(t)}. Illustrate each answer with a suitable waveform sketch. b) State by inspection (i.e. without performing any formal analysis)...
  44. Aleoa

    I Probability density functions for velocity and position

    In the first volume of his lectures (cap. 6-5) Feynman asserts that these 2 can be the PDF of velocity and position of a particle. Under which conditions it's possible to model velocity and position of a particle using these particular PDFs ? ps: Is the "Heisenberg uncertainty principle"...
  45. Z

    Grade 11 Math Help Quadratic functions/ physics

    1. Determine the equation that represents the relationship between the power and the current when the electric potential difference is 24v and the resistance is 1.5 Ω. 2. Draw a graph of the parabola that corresponds to the equation found in (a). 3. Determine the current needed in order for...
  46. T

    Metric space of continuous & bounded functions is complete?

    Homework Statement The book I'm using provided a proof, however I'd like to try my hand on it and I came up with a different argument. I feel that something might be wrong. Proposition: Let ##<X,d>## be a metric space, ##<Y,D>## a complete metric space. Then ##<C(X,Y), \sup D>## is a complete...
  47. Theia

    MHB Numerics on wild oscillating functions

    Hello! I'd like to ask for a help about how to compute accurately functions which has very intense oscillations. My example is to estimate I = \int_0^{\infty} \sin(x^2) dx= \int_0^{\infty}\frac{\sin(t)}{2\sqrt{t}} dt.I tried trapezoid rule over one oscillation at a time, but result is poor. My...
  48. M

    Mathematica Can I Scale Down Basis Functions Without Losing Zero Force?

    Hi PF! I'm working with some basis functions ##\phi_i(x)##, and they get out of control big, approximately ##O(\sinh(12 j))## for the ##jth## function. What I am doing is forcing the functions to zero at approximately 3 and 3.27. I've attached a graph so you can see. Looks good, but in fact...
  49. M

    I Shifting polar functions vertically

    Hi PF! I have a function that looks like this $$f(r,\theta) = \sinh (\omega \log (r))\cos(\omega(\theta - \beta))$$ You'll notice ##f## is harmonic and satisfies the BC's ##f_\theta(\theta = \pm \beta) = 0##. Essentially ##f## has no flux into the wall defined at ##\theta = \pm \beta##. So we...
  50. K

    I Tensors: Bar Symbol Over Functions or Indices?

    When dealing with any tensor quantity, when making a coordinate transformation, we should put a bar (or whatever symbol) over the functions or over the indices? For exemple, should the metric coefficients ##g_{\mu \nu}## be written in another coord sys as ##\bar g_{\mu \nu}## or as ##g_{\bar \mu...
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