What is Periodic functions: Definition and 49 Discussions

A periodic function is a function that repeats its values at regular intervals, for example, the trigonometric functions, which repeat at intervals of 2π radians. Periodic functions are used throughout science to describe oscillations, waves, and other phenomena that exhibit periodicity. Any function that is not periodic is called aperiodic.

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  1. Q

    I Orthogonal Basis of Periodic Functions: Beyond Sines and Cosines

    Hello everyone, I've been delving deep into the realm of periodic functions and their properties. One of the fundamental concepts I've come across is the use of sines and cosines as an orthogonal basis for representing any functions. This is evident in Fourier series expansions, where any...
  2. Infrared

    Challenge Math Challenge - June 2023

    Welcome to the reinstatement of the monthly math challenge threads! Rules: 1. You may use google to look for anything except the actual problems themselves (or very close relatives). 2. Do not cite theorems that trivialize the problem you're solving. 3. Have fun! 1. (solved by...
  3. Ahmed1029

    Sonometers, tuning forks, and wave periodicity

    Summary: Cofnusion regarding waves on a sonometer band A tuning fork is used to determine the wave frequency of a sonometer(according to my understanding), so whay about pulse waves? Does a pulse have a wave frequency? Couldn't a pulse travel over the sonometer band that can be determined by a...
  4. R

    A Measure of non-periodicity of almost periodic functions

    As is well known, almost periodic functions can be represented as a Fourier series with incommensurable (non-multiple) frequencies https://en.wikipedia.org/wiki/Almost_periodic_function. It seems to me that I came up with an integral criterion for the degree of non-periodicity. The integral of a...
  5. R

    Fourier series, periodic function for a string free at each end

    From the statement above, since the ring is massless, there's no force acting vertically on the rings. Thus, the slope is null. ##\frac{\partial y(0,0)}{\partial x} = \frac{\partial y(L,0)}{\partial x} = 0## ##\frac{\partial y(0,0)}{\partial x} = A\frac{2 \pi}{L}cos(\frac{2 \pi 0}{L}) =...
  6. D

    Solution of the f_1(x)-f_1(x-pi)=f_2(x) functional equation

    Laplace transform of eq. [1] [4] F1(p)-exp{-pi*p}*F1(p) = F2(p) Rearranging eq. [4] [5] F1(p) = frac{1}{1-exp{-pi*p}}*F2(p) Inverse LT of eq. [5]
  7. SisypheanZealot

    Dirac Delta using periodic functions

    I know it is something simple that I am missing, but for the life of me I am stuck. So, I used the identity ##sin(a)sin(b)+cos(a)cos(b)=cos(a-b)## which gives me $$\int^{\infty}_{-\infty}dx\:f(x)\delta(x-y)=\int^{\infty}_{-\infty}dx\:f(x)\frac{1}{2L}\sum^{\infty}_{n=-\infty}\lbrace...
  8. S

    Specifying vertical asymptotes in periodic functions in set notation

    Hi all, What is the general set notation for specifying a vertical asymptote and domain for a periodic function? For example, if I have a periodic function which has a period of pi/2, and within that period, a vertical asymptote occurs at pi/4. The domain is R, excluding that vertical...
  9. C

    MHB Anti-derivatives of the periodic functions

    Dear Everyone, I do not know how to begin with the following problem:Suppose that $f$ is $2\pi$-periodic and let $a$ be a fixed real number. Define $F(x)=\int_{a}^{x} f(t)dt$, for all $x$ . Show that $F$ is $2\pi$-periodic if and only if $\int_{0}^{2\pi}f(t)dt=0$. Thanks, Cbarker1
  10. Rectifier

    I Periodic Functions: Meaning of 1-Periodicity

    I know that some functions are ## 2 \pi ## periodic but what does it mean that a function is ##1##-periodic. Is it ##f(x+1n) = f(x)## where ## n \in \mathbb{Z} ## ?
  11. M

    MHB Trigonometry and periodic functions

    !HELP! The Singapore Flyer, until recently the world's largest Ferris wheel, completes one rotation every 32 minutes.1 Measuring 150 m in diameter, the Flyer is set atop a terminal building, with a total height of 165 m from the ground to the top of the wheel. When viewed from Marina Centre, it...
  12. J

    Complex periodic functions in a vector space

    Homework Statement Consider the set V + {all periodic *complex* functions of time t with period 1} Draw two example functions that belong to V. Show that if f(t) and g(t) are members of V then so is f(t) + g(t)Homework EquationsThe Attempt at a Solution f(t) = e(i*w0*t)) g(t) =e(i*w0*t...
  13. TheSodesa

    Amplitude and phase spectra from fundamental frequency?

    Homework Statement Let \begin{equation*} f(t) = 2 + \cos\left( 3t - \frac{\pi}{6} \right) + \frac{1}{4}\cos\left( \frac{1}{2}t + \frac{\pi}{3} \right) + \sin^2(t) \end{equation*} Determine the period ##T## and fundamental frequency ##\omega_0## of ##f## and draw images of its amplitude and...
  14. Joppy

    MHB Fourier Transform of Periodic Functions

    A tad embarrassed to ask, but I've been going in circles for a while! Maybe i'll rubber duck myself out of it. If f(t) = f(t+T) then we can find the Fourier transform of f(t) through a sequence of delta functions located at the harmonics of the fundamental frequency modulated by the Fourier...
  15. Evangeline101

    Number of hours of daylight - Periodic functions.

    Homework Statement Homework Equations none The Attempt at a Solution a) It is a periodic relationship because the number of hours of daylight repeats each year? OR It is a periodic relationship because the number of hours of daylight is based on the rotation of the earth, which is also...
  16. thegreengineer

    I Fourier Series: I don't understand where I am wrong --

    Good afternoon people. Recently I started taking a course at my college about Fourier series but I got extremely confused. Here's what's going on. In school we were asigned to use the symmetry formulas to find the Fourier series of the following: f\left ( t \right )=\begin{cases} 1 & \text{ if...
  17. alexandria

    Periodic Functions Homework: Daylight Hours

    Homework Statement Homework Equations no equations required The Attempt at a Solution [/B] a) The number of hours of daylight is a periodic relationship, because it repeats the same wave-like pattern over the course of 1-2 years. b) the period is the amount of time it takes for one cycle...
  18. U

    Periodic Functions: Find Fundamental Period & Graph Solution

    Homework Statement What is the fundamental period of the expression sinx/sinx.can you guys please illustrate how to make its graph? Homework Equations Okay I know drawing graph can give me the period.Can the period be found by any other method? The Attempt at a Solution I'm told that the...
  19. F

    Is a Bloch wave periodic in reciprocal space?

    A Bloch wave has the following form.. ## \Psi_{nk}(r)=e^{ik\cdot r}u_{nk}(r)## The ##u_{nk}## part is said to be periodic in real space. But what about reciprocal space? I've had a hard time finding a direct answer to this question, but I seem to remember reading somewhere that the entire...
  20. **Mariam**

    Pendulum periodic motion; period parameter

    Homework Statement A pendulum with a mass of 0.1 kg was released. The string made an angle of 7 ° with the vertical. The bob of the pendulum returns to its lowest point every 0.1 seconds. What is the period, frequency? Homework Equations T= 1/f T=sec/cycles F= cycles/sec The Attempt at a...
  21. S

    What is the frequency of the sum of several sine waves?

    I am given three sine waves with individual frequency being 10 Hz, 50 Hz, and 100 Hz. What is the frequency of the following : y(t) = sin(2π10t) + sin(2π50t) + sin(2π100t) Is it simply 100, the LCM of all the sin waves? If not, How to calculate the frequency of y(t) ?
  22. Titan97

    If a function satisfies f(x+1)+f(x-1)=root(2).f(x),....

    Homework Statement If a function satisfies f(x+1)+f(x-1)=√2.f(x), the the period of f(x)=_____ Homework Equations none The Attempt at a Solution I first tried to get rid of the irrational term. f(x+2)+f(x)= √2.f(x+1) f(x)+f(x-2) = √2.f(x-1) adding the above equation, and substituting the...
  23. W

    Periodic Functions: Is Irrationality the Cause of Non-Periodicity?

    Hey. Assume you have a signal ##f## with period ##T_f## and a signal ##g## with period ##T_g##. Then the signal ##h= f+g## is periodic iff ##T_f/T_g \in \mathbb{Q}##. So if ##T_f/T_g## is an irrational number, the signal ##h## will not be periodic. Why is this actually the case?
  24. D

    Sum of Related Periodic Functions

    I have been looking through the book Counterexamples: From Elementary Calculus to the Beginning of Calculus and became interested in the section on periodic functions. I thought of the following question: Suppose you have a periodic real valued function f(x) with a fundamental period T. Let c...
  25. J

    Spatial and temporal periods and periodic functions

    A periodic function is one that ##f(\theta) = f(\theta + nT)##, by definition. However, the argument ##\theta## can be function of space and time ( ##\theta(x, t)## ), so exist 2 lines of development, one spatial and other temporal: $$f(\theta) = f(kx + \varphi) = f(2 \pi \xi x + \varphi) =...
  26. B

    Can Multiple Periods Determine a Fundamental Period in Functions?

    Can a function have two periods? If so, which is the fundamental period? Consider the following function, $$ f : \mathbb{N} → \mathbb{R} $$, defined by f[n] = 1 if n is a multiple of 2 or 3, and 0 otherwise. Then it is clear that 2 and 3 are both periods of this function, since translation...
  27. J

    Representations of periodic functions

    Correct me if I'm wrong, but exist 3 forms for represent periodic functions, by sin/cos, by exp and by abs/arg. I know that given an expression like a cos(θ) + b sin(θ), I can to corvert it in A cos(θ - φ) or A sin(θ + ψ) through of the formulas: A² = a² + b² tan(φ) = b/a sin(φ) = b/A...
  28. E

    Sequences of periodic functions converging to their average value

    Homework Statement Let f be a 2π-periodic function (can be any periodic really, not only 2π), and let g be a smooth function. Then lim_{n\rightarrow∞}\int^{B}_{A} f(nx)g(x) converges to \frac{1}{2π}\int^{2π}_{0}f(x) The Attempt at a Solution So far, I've come up with somewhat of...
  29. M

    MHB Proving Uniqueness of Fourier Coefficients for Continuous Periodic Functions

    Let $f:\mathbb R\to\mathbb R$ be a continuous function of period $2\pi.$ Prove that if $\displaystyle\int_0^{2\pi}f(x)\cos(nx)\,dx=0$ for $n=0,1,\ldots$ and $\displaystyle\int_0^{2\pi}f(x)\sin(nx)\,dx=0$ for $n=1,2,\ldots,$ then $f(x)=0$ for all $x\in\mathbb R.$ I know this has to do with the...
  30. L

    The sum and multiplication of periodic functions

    Homework Statement Hi, my question is whether the sum and multiplication of two periodic functions (with a common period) are periodic. Our functions are R\rightarrowR. Homework Equations The Attempt at a Solution f(x)=f(x+T) g(x)=g(x+T) T is the period. h(x)=f(x)+g(x)...
  31. Z

    Periodic functions in dilations

    is it possible to find a function f different from f(x)=constant with the property f(kx)=f(x) for some real and positive 'k' ? this is somehow 'dilation periodicity' is the equivalent to the periodic funciton f(x+k)=f(x) for some positive 'k' for the traslation group
  32. Z

    Periodic functions (or similar)

    are there non-connstant function that satisfy the following asumptions ?? y(x)=y(kx) they are 'periodic' but under DILATIONS and also satisfy the differential equation of the form (eigenvalue problem) axy'(x)+bx^{2}y''(x)=e_{n}y(x) if the Lie Group is of translations y(x+1)=y(x)...
  33. T

    I need some recommandations for literature about periodic functions

    Hello, I know I am asking for advice about a very specific topic - periodic functions, almost periodic functions and quasi-periodic functions. I was hit by an idea and I need to know a few things more comprehensively about this topic !?~ :] I am aware that "periodic functions etc." isn't a...
  34. A

    Complex notation of periodic functions

    I have found in physics literature a periodic function of time is many times written in complex form. For example,f(x,t)=g(x)e^{i\omega t} As a non-physicist this has proven a bit confusing. Is it generally understood that the function we are really interested in the real part of the...
  35. Z

    Proving Periodicity of an Odd Function with Period p

    Homework Statement If f be a periodic function as well as an odd function with period p and and x belongs to [-p/2, p/2]. Prove that is periodic with period p. The Attempt at a Solution In the solution, there is a step which I did not understand- I see no property of definite...
  36. K

    Is the quotient of periodic functions also periodic?

    Homework Statement Let f be periodic with period P. Prove that 1/f is periodic with period P. The Attempt at a Solution f(s+P)= f(s) I know that is the equation for a periodic function. I am not sure how to prove the 1/f part though. Would I just do this: 1/f(s+P)=f(s)? I'm just...
  37. E

    F(x) = x as a sum of periodic functions?

    someone told me that there's a proof that says f(x) = x can be expressed as a sum of two periodic functions.. does anybody know this? thanks for sharing
  38. J

    Proving sums of periodic functions need not be periodic(almost periodic)

    Homework Statement Hi and thank you for reading this! Let \left.f(x) = cos(x) + cos\left(\pi x\right) a) show that the equation f(x)=2 has a unique solution. b) conclude from part a that f is not periodic. Does this contradict withe the previous exercise that states if...
  39. Q

    How Do You Model Desert Temperature Variations Mathematically?

    Homework Statement The desert temperature, H, oscillates daily between 40 degrees F at 5 am, and 80 degrees F at 5 pm. Write a possible formula for H in terms of t, measured in hours from 5 am. The Attempt at a Solution The best I can come up with is H=60+40sin((pi/6)t) but this...
  40. T

    Cardinality of set of real periodic functions

    what is the cardinality of a set A of real periodic functions ? f(x)=x is periodic so R is subset of A but not equal because sin(x) is in A but not in R. hence aleph_1<|A|.
  41. N

    Understanding Odd, Periodic Functions: Integrals and Periodic Shifts

    Homework Statement Hi all. Can you confirm these statements: 1) If I integrate an odd, periodic function of period 2L over one period, then the integral equals zero. 2) If I have a function f(x) with period 2L, then f(x+alfa), where alfa is an arbitrary number, will not change it's...
  42. N

    Continuity and periodic functions

    Homework Statement We have a piecewise continuous function and T-periodic function f and we have that: F(a) = \int_a^{a + T} {f(x)dx} I have to show that F is diferentiable at a if f is continuous at a. My attempt so far: I have showed that F is continuous for all a. If we look at one...
  43. B

    Notes and soundwaves and periodic functions and algebra

    this has to do with music and notes and soundwaves and stuff i have to find out the sound waves for thirds (eg c and e or d and f or e and g... u get the picture). its in radians by the way. i figured id worked out the formula for c and the formila for e and just add them but I am not sure...
  44. N

    Periodic Functions Homework: Eigenvalues & Oscillations

    Homework Statement What kind of conditions do eigenvalues impose to ensure periodicity? Is it plausible to say that irrational multiples of eigenvalues imply no harmonic oscillations, if so why? Homework Equations The Attempt at a Solution
  45. N

    Solving Fourier Series for Periodic Functions

    I understand what the Fourier Theorem means, as well as how it behaves, I just don't understand how the math actually pans out or in what order to do what. I'm going to start off with what I know. f(x) = \frac{a_0}{2} \sum_{n=1}^{\infty}(a_n}\cos{nx} + b_{n}\sin{nx}) while, a_{0} =...
  46. G

    Periodic Functions: Is There a Non-Trig Function?

    Is there a continuous periodic function which is not trigonometric. if yes, what?
  47. O

    Finding Laplace Transform Limits for Periodic Functions

    How would you go about finding the limits of the general laplace transform function for periodic functions?
  48. S

    Math Practice: Periodic Functions

    I'm doing a practice exam for a math test on thursday, wondering if anyone could help figure out how to get from one step to the next. i don't think that the background info is necessary for these two steps. the file is attached (Adobe acrobat). what i am wondering about is the answer...