Functions Definition and 1000 Threads

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. 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...
  2. 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) /...
  3. 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...
  4. 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...
  5. 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...
  6. 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...
  7. 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...
  8. 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...
  9. 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 +...
  10. 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...
  11. B

    Two Functions with the Same Derivative

    Homework Statement I am trying to prove without using the mean value theorem that two different functions with the same derivative differ only by a constant. Is it possible to do this without the mean value theorem? If so, would someone help guide me towards the right solution. Homework...
  12. G

    Number of injections, of functions

    Homework Statement Show that the number of injections from ##E \rightarrow F ##, ##E,F## finite sets, ## p = \#E ,\ n = \#F,\ p\le n ##, is ##\frac{n!}{(n-p)!} ## Then, find the number of functions from ##E \rightarrow F ##. Homework Equations ## A = \{ \text{injections from }E \rightarrow F...
  13. Coffee_

    Generating functions and Legendre transforms

    I am confused about the ''4 basic types'' of generating functions. I have searched for this a bit on google but haven't found anything that truly made the click for me on this concept so I'll try here: What I do understand and need no elaboration on: 1) When considering the Hamiltonian and...
  14. Feodalherren

    Efficient Matrix Multiplication with Nested Functions for MATLAB Homework

    Homework Statement Homework Equations [/B] ---------------------- function [C]=mymatmult(A,B) [L1 C1]=size(A); [L2 C2]=size(B); if C1 ~= L2 error('dimension mismatch'); end %if ERROR C=zeros(L1,C2); for i=1:L1 for j=1:C2 C(i,j)=A(i,:)*B(:,j); end %in for end %out for end %function...
  15. Chef Hoovisan

    Solving a system of recursive functions

    I've run across a system of recursive functions (call them f and g). The system looks like this: f(x) = a f(x-1) + b g(x-1) g(x) = a g(x-1) + c f(x-1) I also know that f(0) = 0 and g(0)>0. Finally, I know for other reasons that are too complicated to go into here that the system is somehow...
  16. D

    Composite Functions, please confirm

    New to composite functions here. Lesson has been vague and unhelpful.. again. Here is what I've worked on so far but not sure on the last equation in particular, or that I have done my multiplication properly when working with a squared set of brackets, multiplied by an number.. (b and c) Any...
  17. C

    Critical points and of polynomial functions

    Homework Statement A rectangular region of 125,000 sq ft is fenced off. A type of fencing costing $20 per foot was used along the back and front of the region. A fence costing $10 per foot was used for the other sides. What were the dimensions of the region that minimized the cost of the...
  18. Emmanuel_Euler

    Integration and special functions.

    what is the relationship between special functions and integration ? why integral of some function like (sqrt(ln(x)) and (cos(1/x) and more) are entering us to special functions?? PLEASE HELP ME TO UNDERSTAND.
  19. S

    MHB [Limits] Help with Delta-Epsilon Proofs for Multivariable Functions

    Hi guys, just having some confusions on the Delta-Epsilon proofs for multivariable limit functions. here is my question: Apply Delta-Epsilon proof for the Lim (x,y) --> (0,0) of (y^3 + 5x^2y)/(y^2 + 3y^2) to show the limit exists. The part that has me confused is the y to the power of 3, where...
  20. M

    How Are Step Functions Used in Calculating Fourier Transforms?

    Hi. Here is one example from my book. Calculate Fourier transform of signal: Here is solution: We can write x(n) as: , where x1(n) is u(n+N)-u(n-N-1). We can write: (we used that cos(n)=(1/2)*(exp(j*n)+exp(-j*n)). Using properties of Fourier transform of discrete signal: , Fourier...
  21. M

    Can different weights help determine the difference between two functions?

    Hi PF! Can any of you help me determine a good measure for how "different" two functions are from each other? I've thought of using something like ##\int_\Omega (f-g)^2 \, dx##. Can anyone recommend a good technique and direct me to the theory so I can understand it well? Thanks so much! Josh
  22. Shackleford

    Determine whether functions are harmonic

    Homework Statement Determine whether or not the following functions are harmonic: u = z + \bar{z} u = 2z\bar{z} Homework Equations z = u(x,y) + v(x,y)i \bar{z} = u(x,y) - v(x,y)i A function is harmonic if Δu = 0. The Attempt at a Solution Δu = Δz +Δ \bar{z} = u_{xx} + v_{xx} +...
  23. T

    Is the difference of two state functions a state function?

    Hello everybody, For my thermodynamics test I have to tell whether or not a quantity is a state function, which is obviously not all too difficult when regarding entropy, enthalpy etc. on their own. However there are a lot of questions where it is about "H-S" or "G-H". Are these not always...
  24. S

    MHB Write the piecewise function in terms of unit step functions.

    Write the piecewise function \[ f(t) = \begin{cases} 2t, & 0\leq t < 3 \\ 6, & 3 \le t < 5 \\ 2t, & t \ge 5 \\ \end{cases} \] in terms of unit step functions. So here is what i;ve got just guessing , I don't think I'm correct. I really need some help. But I got...
  25. D

    Potential Step and Wave Functions

    Homework Statement Homework EquationsThe Attempt at a Solution For x>b, Ψ(x) = Ae-ikx + Beikx , where k = (√2mE)/hbar a<x<b Ψ(x) = Ce-ik'x + Deik'x , where k = (√2m(U2 - E)/hbar This is the problem part 0<x<a Ψ(x) = Fsink''x...
  26. anemone

    MHB Value of $\dfrac{2k^2}{k-1}$: Solving the Equation

    Determine the value of $\dfrac{2k^2}{k-1}$ given $\dfrac{k^2}{k-1}=k^2-8$.
  27. CAH

    Domain and range of composite functions?

    hey! How do you work out the domain and range of fg(x), do you work out what range of g(x) will fit the domain of f(x)? I have no clue. Thanks
  28. J

    What makes these initial functions so special?

    People say that if you could break a function down into these three functions (constant, successor, projection or sometimes called initial/basic functions) using some operators, then it is primitive recursive. What makes these three functions so special?
  29. Q

    MHB Exponential and Logarithmic Functions

    y=CektA) First find k. [Hint:Use the given information of y=100 when t=2, and y=300 when t=4 to compute k.] B) Finally, find the value for C. [Hint use ine of the two pieces of information given in the problem to solve for C. in other words, use either y=100 when t=2 or use y=300 when 4=4 to...
  30. K

    How do you integrate dirac delta functions?

    Homework Statement ∫δ(x3 - 4x2- 7x +10)dx. Between ±∞. Homework EquationsThe Attempt at a Solution Well I don't really know how to attempt this. In the case where inside the delta function there is simply 2x, or 5x, I know the answer would be 1/2 or 1/5. Or for say δ(x^2-5), the answer would...
  31. Cookiey

    Inverse Function of Greatest Integer Function in a Given Domain

    Homework Statement If f:(2,4)-->(1,3) where f(x)=x-[x/2] (where[.] denotes the greatest integer function), then find the inverse function of f(x). Homework Equations (None I believe.) The Attempt at a Solution I know that for a function to be invertible, it must be both one-one and onto...
  32. R

    Questions related to Relations and Functions

    Homework Statement 1. Range of the function ## \sqrt {x^2+x+1} ## is equal to? 2.ƒ:R---->R is defined as ƒ(x) = x2 -3x +4, then f -1 (2) is equal to?Homework Equations NA The Attempt at a Solution For the first one tried squaring on both the sides but that does not give linear x in terms of...
  33. ElijahRockers

    Convolution of gaussian functions

    Homework Statement Recall that we have defined the Gaussian ##f_s## by ##f_s (t) = \sqrt{s}e^{-st^2}## and shown that ##\hat{f_s}(\lambda) = \frac{1}{\sqrt{2}}e^{\frac{-\lambda^2}{4s}}##. Show that ##f_3 \ast f_6 (t) = \sqrt{\pi}f_{1/2}(t) = \sqrt{\pi/2}e^{-t^{2}/2}## The Attempt at a...
  34. S

    Finding Function Values on a Graph: f(30) and f(-14) Explained

    1. Homework Statement A graph of y=f(x) is shown. Find the following function values and justify your answers. f(30)= f(-14)= Homework EquationsThe Attempt at a Solution I know the graph is periodic, I know it's max and min, and I know it's amplitude because of that. But I don't know what...
  35. D

    Trigonomic Functions, Plotting from equation

    Hi All, Having a tough time with this one and I'm not sure why. Need to state amplitude, period and phase shift of f(x)=3cos2[x-(π/4)]+1. Amplitude being 3, period being 2π/2=π and phase shifted (π/4) to the right. Midline would also be at y=1 Good so far? Right, so I know that 1/4 phase...
  36. I

    First law of thermodynamics & state functions

    Homework Statement 1 kg air at the pressure ##10^6##Pa and the temperature ##125^\circ C = 398K## expand until the volume is 5 times larger. The expansion is done with change in heat at every moment being ##1/4## of the work done by the gas. Calculate the end pressure.Homework Equations ##dU...
  37. C

    MHB Surjective functions from a set of size n+3 to a size of n

    Hello, I wonder if anyone could settle a disagreement I'm having with one of my peers. The question is 'How many surjective functions are there from a set of size n+3 to a set of size n?'. Now, I've already proven that there are (n+1 choose 2)n! surjective functions from a set of size n+1 to a...
  38. T

    Domain and range of multivariable functions

    Homework Statement Specify the domain and range of f(x, y) = arccos(y − x2). Indicate whether the domain is (i) open or closed, and (ii) bounded or unbounded. Give a clear reason in each case.Homework EquationsThe Attempt at a Solution y-x2 >= -1 y >= x2 -1 y-x2 <= 1 y <= x2 +1 I sketched it...
  39. E

    MHB Working with Piecewise Functions

    I'm given the following Piecewise function when $f:[0,1]\to[0,1]$: $f(x) = x$ when $x\in\Bbb{Q}$ $f(x) = 1-x$ when $x\notin\Bbb{Q}$ I need to prove that $f$ is continuous only at the point $x=\frac{1}{2}$. For this problem, I know I need to use the fact that a function $f$ is continuous at a...
  40. kostoglotov

    Q about 2nd derivative test for multivariable functions

    Homework Statement So the test is to take the determinant (D) of the Hessian matrix of your multivar function. Then if D>0 & fxx>0 it's a min point, if D>0 & fxx<0 it's a max point. For D<0 it's a saddle point, and D=0 gives no information. My question is, what happens if fxx=0? Is that...
  41. AdityaDev

    Proving sinx+cosx is not one-one in [0,π/2]

    Homework Statement Prove that sinx+cosx is not one-one in [0,π/2] Homework Equations None The Attempt at a Solution Let f(α)=f(β) Then sinα+cosα=sinβ+cosβ => √2sin(α+π/4)=√2sin(β+π/4) => α=β so it has to be one-one [/B]
  42. C

    MHB Injective and surjective functions

    Hello, I've been reading about injectivity from Z to N and surjectivity from N to Z and was wondering whether there was some kind of algorithm that could generate these specific types of functions?
  43. F

    How to prove some functions are scalar field or vector field

    Homework Statement Homework EquationsThe Attempt at a Solution I solved #2,4 but I don't understand what #1,3 need to me. I know that scalar field is a function of points associating scalar value. But how can I prove some function is scalar field or vector field?
  44. DrPapper

    Exploring Mary Boas' Theorem III: Analytic Functions & Taylor Series

    On page 671 Mary Boas has her Theorem III for that chapter. Roughly it tells us that if f(z) -a complex function- is analytic in a region, inside that region f(z) has derivatives of all orders. We can also expand this function in a taylor series. I get the part about a Taylor series, that's...
  45. ellipsis

    [Algebra] Proving equations involving modulo functions.

    I would like to know some general properties of the modulo (remainder) function that I can use to rewrite expressions. For example, say we wanted to prove the following by rewriting the right-hand-side: $$ \Big{\lfloor} \frac{n}{d} \Big{\rfloor} = \frac{n - n \pmod d}{d} $$ I have no idea how...
  46. Khronos

    Optimisation - Critical Numbers for Complex Functions.

    Hi everyone, I need a little bit of help with an optimization problem and finding the critical numbers. The question is a follows: Question: Between 0°C and 30°C, the volume V ( in cubic centimeters) of 1 kg of water at a temperature T is given approximately by the formula: V = 999.87 −...
  47. N

    Einstein's Field Equations: Effective Potential Functions

    I have seen written out in various places (including this forum) the effective potential function that comes from the solutions to the Schwarszschild Geodesic. But I haven't been able to find the effective potential functions for other solutions to Einstein's field equations. Are there...
  48. nuclearhead

    What functions of fields describe particles?

    I was thinking about the connection between fields and particles. For instance the scalar field Φ(x) and the field Φ(x)+a both represent the same scalar particle. Because the action ∫∂Φ∂Φdx^4 is unaltered and the propagator <0|[Φ(x)+a,Φ(y)+a]|0> is presumably the same. What about if we replace...
  49. M

    MHB Yes, $2^{2^{2^n}}$ is a good example for $p(n)$ and $q(n)$ could be $n!$.

    Hey! :o Find an order $f_1, f_2, \dots f_{30}$ of the functions that satisfies the relations $f_1=\Omega(f_2), f_2=\Omega(f_3), \dots, f_{29}=\Omega(f_{30})$$$\frac{n}{\lg n} , \ \ n^{\lg n} ,\ \ (\sqrt{2})^{\lg n}, \ \ n^2, \ \ n!, \ \ (\lg n)! ,\ \ \left( \frac{3}{2} \right)^n ,\ \ n^3 ,\ \...
  50. G

    What Is the Correct Calculation for 643 + 364 Using Defined Functions?

    Homework Statement For positive integers m, k, and n , let mkn be defined as mkn = kmn , where k\frac {m}{n} is a mixed fraction. What is the value of 643 + 364 ? Homework Equations I attempt the other few similar questions where the solution are as follow 832 + 382 = \frac {169}{24} 641 +...
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