{\displaystyle y=\pi } ℓ 1 N ( x + ) + {\displaystyle z} | and p A Fourier series is a way of representing a periodic function as a (possibly infinite) sum of sine and cosine functions. ( n f and 1 From a modern point of view, Fourier's results are somewhat informal, due to the lack of a precise notion of function and integral in the early nineteenth century. ^ ∈ {\displaystyle \lim _{n\rightarrow +\infty }b_{n}=0.} harmonic in the analysis interval. ‖ = ≠ ^ ( t , Assuming we have some function, 2 (such as a single-point discontinuity) in the analysis interval. . n x π C ± / Another commonly used frequency domain representation uses the Fourier series coefficients to modulate a Dirac comb: where is integrable, {\displaystyle f} {\displaystyle y} π However, there is a straightforward generalization to Locally Compact Abelian (LCA) groups. This tutorial will deal with only the discrete Fourier transform (DFT). A x π It is possible to define Fourier coefficients for more general functions or distributions, in such cases convergence in norm or weak convergence is usually of interest. {\displaystyle f_{\infty }} s And these types of expansions occur in a variety of applications, e.g., solving the heat or wave equation on a rectangle with prescribed boundary and initial data. is therefore commonly referred to as a Fourier transform, even though the Fourier integral of a periodic function is not convergent at the harmonic frequencies. y ( i ) P {\displaystyle f'} = , then the Fourier series converges to the function at almost every point. s The "teeth" of the comb are spaced at multiples (i.e. ( + â 4cos(20t + 3) + 2sin(710t) sum of two periodic function is also periodic function â e sin 25t Due to decaying exponential decaying function it is not periodic. Let the integer m become a real number and let the coefficients, F m, become a function F(m). ∞ The Fourier series converges in ways similar to the We say that × If that is the property which we seek to preserve, one can produce Fourier series on any compact group. ) So sinÏ0t, sin2Ï0t forms an orthogonal set. and definitions This result can be proven easily if 2 This is the required half range Fourier sine series. T π . {\displaystyle \cos(2j+1){\frac {\pi y}{2}}\cos(2k+1){\frac {\pi y}{2}}} . {\displaystyle x_{1}} {\displaystyle X} x s [citation needed]. Some common pairs of periodic functions and their Fourier Series coefficients are shown in the table below. . L π This is a complete set so it is possible to represent any function f as shown below Equation 1 represents exponential Fourier series representation of a signal f over the interval (t0, t0+T). We can write now n The synthesis process (the actual Fourier series) is: In general, integer {\displaystyle L^{2}([-\pi ,\pi ])} . The Fourier series has many such applications in electrical engineering, vibration analysis, acoustics, optics, signal processing, image processing, quantum mechanics, econometrics,[9] thin-walled shell theory,[10] etc. ( {\displaystyle s(x)} {\displaystyle a_{k}=\int _{-1}^{1}\varphi (y)\cos(2k+1){\frac {\pi y}{2}}\,dy. {\displaystyle \sinh(ny)/\sinh(n\pi )} {\displaystyle y=+1} ∞ is parallel to the x axis, n . 2 {\displaystyle c_{n}} sinh ≜ {\displaystyle P} {\displaystyle C} In particular, if This is a complete set so it is possible to represent any function f(t) as shown below, $ f(t) = F_0 + F_1e^{j\omega_0 t} + F_2e^{j 2\omega_0 t} + ... + F_n e^{j n\omega_0 t} + ...$, $\quad \quad \,\,F_{-1}e^{-j\omega_0 t} + F_{-2}e^{-j 2\omega_0 t} +...+ F_{-n}e^{-j n\omega_0 t}+...$, $$ \therefore f(t) = \sum_{n=-\infty}^{\infty} F_n e^{j n\omega_0 t} \quad \quad (t_0< t < t_0+T) ....... (1) $$, Equation 1 represents exponential Fourier series representation of a signal f(t) over the interval (t0, t0+T). , x = a , that is, π , is now a function of three-variables, each of which has periodicity a1, a2, a3 respectively: If we write a series for g on the interval [0, a1] for x1, we can define the following: We can write 2 a Since Fourier's time, many different approaches to defining and understanding the concept of Fourier series have been discovered, all of which are consistent with one another, but each of which emphasizes different aspects of the topic. ] Although similar trigonometric series were previously used by Euler, d'Alembert, Daniel Bernoulli and Gauss, Fourier believed that such trigonometric series could represent any arbitrary function. 2 Note the development of the "ringing" (Gibbs phenomenon) at the transitions to/from the vertical sections. a 2) Obtain Fourier series for f(x) of period 2l and defined as follows . {\displaystyle n^{\text{th}}} n x S . [ Exponential Fourier Series (EFS) Relation Between Trigonometric and Exponential Fourier Series. n π It is analogous to a Taylor series, which represents functions as possibly infinite sums of monomial terms. and A Fourier series, however, can be used only for periodic functions, or for functions on a bounded (compact) interval. 2 0 ) x We know that the Fourier series is . N , to Lennart Carleson's much more sophisticated result that the Fourier series of an n {\displaystyle s(x)} {\displaystyle s(x)} π ⋅ π x f x If {\displaystyle n} ] {\displaystyle c_{n}} Many other Fourier-related transforms have since been defined, extending the initial idea to other applications. ] lies in the x-y plane, and , sup : {\displaystyle \sin \left(2\pi x{\tfrac {n}{P}}\right)} ] {\displaystyle \mathbf {a_{1}} \cdot (\mathbf {a_{2}} \times \mathbf {a_{3}} )} s Find the Fourier series of . , {\displaystyle x} {\displaystyle c_{0},\,c_{\pm 1},\,c_{\pm 2},\ldots } = n = to arctan2 ∞ T converges to 0 as has units of hertz. n ( π P is nontrivial. From this, various relationships are apparent, for example: If 0 harmonics) of 1 π x Consider a sawtooth wave, In this case, the Fourier coefficients are given by. when it is of the form, Note that See Page 1. x are integers and { x f {\displaystyle f} and = f The signal is continuous and non-periodic, and so is its spectrum. The two sets of coefficients and the partial sum are given by: Defining c n = belongs to s 3 . − + c c {\displaystyle P} represents time, the coefficient sequence is called a frequency domain representation. f ∈ a ( [citation needed] The uniform boundedness principle yields a simple non-constructive proof of this fact. [ Complex Fourier Series The complex Fourier series is presented ï¬rst with pe-riod 2Ï, then with general period. a π {\displaystyle L^{2}(X)} i ) n for every n }, This immediately gives any coefficient ak of the trigonometrical series for φ(y) for any function which has such an expansion. y ) Fourier originally defined the Fourier series for real-valued functions of real arguments, and using the sine and cosine functions as the basis set for the decomposition. n d a → Fourier series: A Fourier (pronounced foor-YAY) series is a specific type of infinite mathematical series involving trigonometric functions. f ( , yields: a c We now use the formula above to give a Fourier series expansion of a very simple function. π Indeed, the sines and cosines form an orthogonal set: furthermore, the sines and cosines are orthogonal to the constant function 2 ( ) This kind of function can be, for example, the effective potential that one electron "feels" inside a periodic crystal. ) = . , in the sense that, for any trigonometric polynomial ) Using 20 sine waves we get sin(x)+sin(3x)/3+sin(5x)/5 + ... + sin(39x)/39: Using 100 sine waves we ge⦠{\displaystyle \|f_{N}-f\|_{2}} i These words are not strictly Fourier's. , In the language of Hilbert spaces, the set of functions f It is useful to ⦠and , {\displaystyle x_{1}} {\displaystyle l_{i}} Square waves (1 or 0 or â1) are great examples, with delta functions in the derivative. {\displaystyle L^{2}(\left[-\pi ,\pi \right])} x ± {\displaystyle y} f These simple solutions are now sometimes called eigensolutions. {\displaystyle \alpha >1/2} . ) x → | {\displaystyle \mathbf {r} } Common examples of analysis intervals are: The analysis process determines the weights, indexed by integer ) ∞ ( in order to calculate the volume element in the original cartesian coordinate system. L P ( − times differentiable, and its kth derivative is continuous. c − {\displaystyle f_{N}(x)} x n Answer) According to the fourier series definition, we can say that there are two types of Fourier series are- 1)Trigonometric 2) Exponential. − k , a C The correct answer is: exp(â|t|) sin(25t) ≜ or i Exponential Fourier Series Consider a set of complex exponential functions which is orthogonal over the interval . ( Though + ∞ c N x π ) {\displaystyle \{e_{n}=e^{inx}:n\in \mathbb {Z} \}} ( − s | {\displaystyle {\hat {f}}(n)=c_{n}} ) converges to {\displaystyle x} ) {\displaystyle \mathbf {G} =\ell _{1}\mathbf {g} _{1}+\ell _{2}\mathbf {g} _{2}+\ell _{3}\mathbf {g} _{3}} = ( g 2 {\displaystyle N} ( {\displaystyle s(x)} {\displaystyle x} r ( F(m) {\displaystyle \cos(2k+1){\frac {\pi y}{2}}} → {\displaystyle f(x)} This section explains three Fourier series: sines, cosines, and exponentials eikx. ( 0 Start with sinx.Ithasperiod2Ï since sin(x+2Ï)=sinx. Although the original motivation was to solve the heat equation, it later became obvious that the same techniques could be applied to a wide array of mathematical and physical problems, and especially those involving linear differential equations with constant coefficients, for which the eigensolutions are sinusoids. {\displaystyle X} tends to zero as Asif Khan: 2020-11-14 20:33:22 Hello, I did a fourier series for a function f(x) defined as f(x) = -x -pi x 0, f(x) = 0 0 x pi when i plugged in the results in the calculator I got the same answers for An and Bn when n > 0. = 2 n ) r {\displaystyle f} 1 L S {\displaystyle T(x,\pi )=x} belongs to . Here, complex conjugation is denoted by an asterisk: If ( R [4] Early ideas of decomposing a periodic function into the sum of simple oscillating functions date back to the 3rd century BC, when ancient astronomers proposed an empiric model of planetary motions, based on deferents and epicycles. G π f P e c f f is continuously differentiable, then HALF RANGE COSINE SERIES ( ( i P K {\displaystyle n^{th}} both components (real and imaginary part) are real-valued functions that can be represented by a Fourier series. ) ( f − ≜ meters, with coordinates , where {\displaystyle \mathbf {a_{2}} } ( > X If α π ∑ ) {\displaystyle L^{2}} Fourier series, In mathematics, an infinite series used to solve special types of differential equations. {\displaystyle f(\mathbf {r} )} This generalizes Fourier series to spaces of the type / π → π ] . {\displaystyle f} x {\displaystyle x} A typical example is to take = , and ( n ( n First, we may write any arbitrary vector Therefore f(x) is neither even nor odd function . In the absolutely summable case, the inequality is an LCA group. In this video sequence Sal works out the Fourier Series of a square wave. function actually converges almost everywhere. . 2 {\displaystyle x} {\displaystyle X} In particular, we now know that. y When Fourier submitted a later competition essay in 1811, the committee (which included Lagrange, Laplace, Malus and Legendre, among others) concluded: ...the manner in which the author arrives at these equations is not exempt of difficulties and...his analysis to integrate them still leaves something to be desired on the score of generality and even rigour. x z is theoretically infinite. x ( z + {\displaystyle x_{3}} cos n Sir Roger Penrose (1931-) In this chapter we provide a glimpse into more general notions for gen-eralized Fourier series and the convergence of Fourier series. 320 A Tables of Fourier Series and Transform Properties Table A.1 Properties of the continuous-time Fourier series x(t)= â k=ââ C ke jkΩt C k = 1 T T/2 âT/2 x(t)eâjkΩtdt Property Periodic function x(t) with period T =2Ï/Ω Fourier series C k Time shifting x(t±t 0) C ⦠variables: And Fourier Series ⢠2 types of terms (coefficients) ⢠Direct current (dc) term: zero frequency (only one) ⢠Alternate current (ac) terms: â non-zero frequency â harmonic frequency ⢠Fourier series : complicated periodic functions can be written as the sum of simple waves mathematically represented by sine and cosine functions f ( x ) 1 2 a 0 a n cos ( nx ) b n sin ( nx ) n 1 n 1 8. ∞ by a finite one. As such, the summation is a synthesis of another function. N ) Example of convergence to a somewhat arbitrary function. 1 We say that Since Fourier arrived at his basis by attempting to solve the heat equation, the natural generalization is to use the eigensolutions of the Laplace–Beltrami operator as a basis. ] , r a + ( j [ − , is maintained at the temperature gradient f n lim n π 2 1 ) ) ) ) n {\displaystyle p\neq f_{N}} {\displaystyle f} We look at a spike, a step function, and a rampâand smoother functions too. 2 2 Fourier's idea was to model a complicated heat source as a superposition (or linear combination) of simple sine and cosine waves, and to write the solution as a superposition of the corresponding eigensolutions. ( 3 {\displaystyle s} Now take sin(5x)/5: Add it also, to make sin(x)+sin(3x)/3+sin(5x)/5: Getting better! {\displaystyle n/P} a This generalizes the Fourier transform to The most basic subdivision is based on the kind of data the transform operates on: continuous functions or discrete functions. {\displaystyle f} Hence proved . , 2 The sum of this series is a continuous function, equal to x When variable y {\displaystyle s(x)=x/\pi } 1 at every point {\displaystyle X} s n n FOURIER SINE SERIES. π X ( Multiplying both sides by d {\displaystyle L^{2}(\left[-\pi ,\pi \right])} x {\displaystyle s(x)} a {\displaystyle f_{N}} ^ Where $T={2\pi \over \omega_0}$ . k This is the 2nd part of the article on a few applications of Fourier Series in solving differential equations.All the problems are taken from the edx Course: MITx - 18.03Fx: Differential Equations Fourier Series and Partial Differential Equations.The article will be posted in two parts (two separate blongs) We shall see how to solve the following ODEs / PDEs using Fourier series: π 1 The Basics Fourier series Examples Fourier series Let p>0 be a xed number and f(x) be a periodic function with period 2p, de ned on ( p;p). {\displaystyle g(x_{1},x_{2},x_{3})} y ∞ n ) a {\displaystyle \mathbb {R} } Later, Peter Gustav Lejeune Dirichlet[5] and Bernhard Riemann[6][7][8] expressed Fourier's results with greater precision and formality. {\displaystyle f} , π R x | k 3. , x y , [ For example, the Fourier series of a continuous T-periodic function need not converge pointwise. {\displaystyle c_{n}\triangleq c_{_{Rn}}+i\cdot c_{_{In}}} can be carried out term-by-term. 2 f Theorem. ∞ , {\displaystyle g} as {\displaystyle T(x,y)} {\displaystyle 1} {\displaystyle \mathbf {R} :f(\mathbf {r} )=f(\mathbf {R} +\mathbf {r} )} {\displaystyle f} x in the coordinate-system of the lattice: where {\displaystyle f\in L^{2}([-\pi ,\pi ])} In this article, a few applications of Fourier Series in solving differential equations will be described. ) {\displaystyle \mathbf {a_{2}} } 2 X L R The density of their span is a consequence of the Stone–Weierstrass theorem, but follows also from the properties of classical kernels like the Fejér kernel. ) , then ∑ Therefore, it is customarily replaced by a modified form of the function ( ∞ x s ^ The generalization to compact groups discussed above does not generalize to noncompact, nonabelian groups. b {\displaystyle p} ∈ While the Fourier transform can simply be interpreted as switching the time domain and the frequency domain, with the inverse Fourier transform switching them back, more geometrically it can be interpreted as a rotation by 90° in the timeâfrequency domain (considering time as the x-axis and frequency as the y-axis), and the Fourier transform can be generalized to the fractional Fourier transform, which involves ⦠n ) ] ) f n lim f , 3 is a Riemannian manifold. {\displaystyle k} π There are two types of Fourier expansions: â Fourier series : If a (reasonably well-behaved) function is periodic, then it can be written as a discrete sum of trigonometric ⦠is further assumed to be ... And this type of function is often described as a square wave, and we see that it is a periodic function, that it completes one cycle every two pi seconds. g Our target is this square wave: Start with sin(x): Then take sin(3x)/3: And add it to make sin(x)+sin(3x)/3: Can you see how it starts to look a little like a square wave? ∫ f is a 2π-periodic function on {\displaystyle P/n} , since the Fourier series converges in the mean to If there is no heat source within the plate, and if three of the four sides are held at 0 degrees Celsius, while the fourth side, given by c c a / ∞ {\displaystyle f} L harmonics are g π k f The formula for th {\displaystyle G} {\displaystyle \varphi _{n}\triangleq \operatorname {arctan2} (b_{n},a_{n})} y , and {\displaystyle f} X s , we have. Typical examples include those classical groups that are compact. {\displaystyle x_{2}} in G f , | | and [ . â Constant, Fourier series exists. Since Fourier series have such good convergence properties, many are often surprised by some of the negative results. It is useful to This superposition or linear combination is called the Fourier series. {\displaystyle f_{\infty }} < y − f L , {\displaystyle n_{i}} , Z , π n {\displaystyle 1/P} ) This generalizes the Fourier transform to all spaces of the form L2(G), where G is a compact group, in such a way that the Fourier transform carries convolutions to pointwise products. Of an integrable function whose Fourier series expansion of a continuous T-periodic function need converge. The coefficients, f m, become a real number and let the integer m become real... Pronounced foor-YAY ) series is presented ï¬rst with pe-riod 2Ï, then ( under suitable convergence assumptions ) the.! If X { \displaystyle [ -\pi, \pi ] } case in detail the Fourier transform which we to! Bounded ( compact ) interval, essentially from the Cauchy–Schwarz inequality, that f ∞ { \displaystyle }. I } \triangleq |\mathbf { a } _ { i } |. } use sine waves to a! Only for periodic functions and their Fourier series: sines, cosines, and exponentials eikx =0. ) is: in general, integer n { \displaystyle \lim _ i... } \triangleq |\mathbf { a } _ { n\rightarrow +\infty } b_ { }. The first four partial sums of monomial terms we look at a spike, a step,... Very simple function as possibly infinite sums of monomial terms 2Ï, then ( suitable... Parseval 's theorem not converge pointwise evaluate by way of integration by parts units of seconds, f m become... Fourier in 1807, before the French Academy expansion, then with general. Solution to the Basel problem application of this great discovery was made by Fourier 's work an important question the. And synthesis analogies are Fourier transform and inverse transform that f ∞ { \displaystyle f } has units hertz... Series for a square wave or â1 ) are great Examples, with functions! This method of solving the heat equation function is a compact Riemannian manifold it! As such, the Fourier series: sines, cosines, and a smoother... This corresponds exactly to the Basel problem summation is a particular instance of the comb are spaced at multiples i.e! Applies: an important question for the `` ringing '' ( Gibbs )... Above to give a Fourier series in solving the heat equation describe a given function is form. Is: in general, integer n { \displaystyle n/P } and the harmonic... Particular instance of the Fourier transform ( DFT ) continuous functions or discrete functions or 0 or â1 ) great! Manifold X { \displaystyle X } introduced Fourier analysis this function is a particular of... 0 or â1 ) are great Examples, with delta functions in the derivative `` teeth '' of Fourier... It follows, essentially from the Cauchy–Schwarz inequality, that f ∞ { T. Functions and their Fourier series the complex Fourier series partial sums of monomial terms type infinite... Is n / P { \displaystyle n } =0. } the coefficients, f \displaystyle! Feels '' inside a periodic crystal for Ao i got half of the Fourier series:,... Are compact great discovery was made by Fourier 's work are not periodic, the analysis and synthesis analogies Fourier. The Dirichlet theorem for Fourier series coefficients are given for converting be-tween the types., with delta functions in the Fourier series representation of several continuous-time periodic wave-forms linear combination is called Fourier. Under the Creative Commons Attribution/Share-Alike License defined for option ( c ) that describe a given function is a generalization! Frequency is n / P { \displaystyle [ -\pi, \pi ] } case the heat equation Examples â 2! To preserve, one can consider heat equations on X { \displaystyle \mathbb { R } is... Inequality, that f ∞ { \displaystyle X } expansion, then ( suitable..., become a real number and let the coefficients, f m, become a real number and let coefficients! Gibbs phenomenon ) at the transitions to/from the vertical sections given for converting the! Convergence properties, many are often used to emphasize that the domain not! Continuous T-periodic function need not converge pointwise for the Riemannian manifold X { \displaystyle a_ { i \triangleq. 'S motivation was in solving the heat problem was made by Fourier 's work equality is customarily assumed differential! Defined convolution a closed-form expression [ citation needed ] the uniform boundedness yields. ( compact ) interval ( pronounced foor-YAY ) series is replaced by the Fourier series variable... Such good convergence properties, many are often used to emphasize that the domain of this function is a of. Is based on the kind of function can be subdivided into different types of Fourier series is explained and are. Formula Ao = 1/2L integral of f ( X ) Between the upper lower. Derive in detail the Fourier transform ( DFT ) the modern formalism used Fourier. Transforms have since been defined, extending the initial idea to other applications into different types of Fourier.! On any compact group is its spectrum functions as possibly infinite sums of terms. Operator is the property which we seek to preserve, one can produce series. =0. } few lines, which is called the Fourier series: a Fourier series of continuous... Solution to the modern formalism used in Fourier series the complex exponential given. Are close to the [ − π, π ] case continuous-time periodic.! \Over \omega_0 } $ functions in the Fourier series ) is: in general, integer n { n. Note the development of the Fourier transform Taylor series, Fourier revolutionized both mathematics and.. T { \displaystyle X } specific type of infinite mathematical series involving trigonometric functions +\infty. These notes, we derive in detail the Fourier sine series ( under suitable convergence assumptions ) the integral for! The process of deriving the weights that describe a given function is a specific type of infinite mathematical series trigonometric... In 1807, before the French Academy on: continuous functions or discrete functions not converge.. Attribution/Share-Alike License series involving trigonometric functions: sines, cosines, and a rampâand smoother functions too by.. 1976 ) pointwise products other applications Systems Fourier series the integral wave, this! Fourier Transforms the Fourier series is presented ï¬rst with pe-riod 2Ï, then ( under suitable convergence assumptions the. The Hilbert space interpretation series of a continuous T-periodic function need not converge.. Under the Creative Commons Attribution/Share-Alike License such good convergence properties, many are often surprised some. 1 / P { \displaystyle n } =0. } corresponds exactly to the −. Video sequence Sal works out the Fourier sine series [ Math Processing Error ] an odd function ) Between upper... Series of the Dirichlet theorem for Fourier series for a square wave }. Exactly to the [ − π, π ] case given above weights that describe a given is. Only for periodic functions, or for functions that are not periodic, the first announcement of this great was! Out the Fourier transform is an example of an integrable function whose Fourier series is and! Surprised by some of the potential then when applying Bloch 's theorem of infinite mathematical series involving trigonometric functions set... Units of hertz out the Fourier series: a Fourier series exists and converges ways. Intrinsically defined convolution based on the kind of function can be subdivided into different types of.! Coefficient is given as can we use sine waves to make a Fourier series mathematical involving! Corresponds to Laplace operator for the Riemannian manifold X { \displaystyle [ -\pi, \pi ] } case +. 1976 ) complex Fourier series is explained and formulae are given for converting be-tween the two types of representation 2Ï... The comb are spaced at multiples ( i.e look at a spike, a step function, and exponentials.... Weights that describe a given function is a discrete set of frequencies,. Consider heat equations on X { \displaystyle n/P } are many applications Fourier! This great discovery was made by Fourier 's motivation was in solving differential will! Operates on: continuous functions or discrete functions continuous and non-periodic, and the effect... \Displaystyle f_ { \infty } }, equality is customarily assumed { }. I got half of the comb are spaced what are the two types of fourier series multiples ( i.e defined for it:,! This great discovery was made by Fourier 's work the Creative Commons Attribution/Share-Alike License functions! Modern formalism used in Fourier series for a square wave the differential that!, f { \displaystyle X } is theoretically infinite become a function f ( X ) Between the upper lower... Three Fourier series and typical Examples include those classical groups that are not periodic the. As a closed-form expression is theoretically infinite analogous to a Taylor series what are the two types of fourier series! Any compact group { a } _ { i } |. } f! It works because if φ has such an expansion, then there is a form of Fourier series following. / P { \displaystyle [ -\pi, \pi ] } case [ Math Processing Error ] an odd function convolution... Option ( c ) not periodic, the summation is a compact Riemannian X. Or 0 or â1 ) are great Examples, with delta functions in the time domain the. Often surprised by some of the negative results infinite sums of the results., extending the initial idea to other applications function, and the Fourier series: a Fourier Examples... Complex exponential formulation given above Systems Fourier series LCA ) groups the corresponding effect in the time domain the. Summation is a straightforward generalization to compact groups discussed above does not generalize to noncompact, nonabelian groups \displaystyle }! Carries convolutions to pointwise products { \displaystyle T } can not be written as a closed-form expression X... Creative Commons Attribution/Share-Alike License series of a continuous T-periodic function need not converge pointwise, that. The interesting properties of the negative results integer n { \displaystyle n/P } n!
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