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# Fourier coefficients

### TheFourierTransform

1. It looks like the whole Fourier Series concept is working. Here is a 7-term expansion (a0, b1, b3, b5, b7, b9, b11): Figure 5. The square waveform and the seven term expansion. The most important equation of this page is Equation 7 - the formulas for the Fourier Series coefficients. These equations give the optimal values for any periodic function
2. ing the coefficients
3. ary investigations by Leonhard Euler, Jean le Rond d'Alembert, and Daniel Bernoulli. Fourier introduced the series for the purpose of solving the heat equation in a metal plate, publishing his initial results in his 1807.
4. It follows that f^(x) has a Fourier series: f^(x+) + f^(x ) 2 = a 0 + X1 n=1 (a n cos(nx) + b n sin(nx)): Since f(x) = f^(Ëx=p), we nd that f also has a Fourier series: f(x+) + f(x ) 2 = a 0 + X1 n=1 a n cos nËx p + b n sin nËx p : Daileda Fourier Coe cient
5. aries: 1. Fourier Series: For a given periodic function of period P, the Fourier series is an expansion with sinusoidal bases having periods, P/n, n=1, 2, p lus a constant. Given: f (t), such that f (t +P) =f (t) then, with P Ï=2Ï, we expand f (t) as a Fourier series by ( ) (
6. The cn and í ĩížn (or Ïn) values can be derived from the an and bn coefficients. To illustrate the significance of the Fourier series decomposition, consider the square wave of Figure 5 (a), which is an even function. Only even function (cosine) terms are non-zero. The first six non-zero Fourier series terms are shown in Figure 5 (b)
7. Notice the subtle difference: we're not trying to show that any is expressible as a fourier sum; we are given one that is, and we merely want to find the right coefficients to properly give us. Massaging into a better form. There are many ways to express a sum of sinusoids. The one we wrote above - a sum of sines with of differing amplitudes and phases - is the easiest to intuitively.

Als Fourierreihe, nach Joseph Fourier (1768-1830), bezeichnet man die Reihenentwicklung einer periodischen, abschnittsweise stetigen Funktion in eine Funktionenreihe aus Sinus - und Kosinusfunktionen. Die Basisfunktionen der Fourierreihe bilden ein bekanntes Beispiel fÃžr eine Orthonormalbasis Coefficients de Fourier. La dÃĐfinition des coefficients de Fourier porte sur les fonctions pÃĐriodiques intÃĐgrables au sens de Lebesgue sur une pÃĐriode. Pour une fonction pÃĐriodique, ÃŠtre de classe L p implique l'intÃĐgrabilitÃĐ. Ceci comprend en particulier les fonctions continues, ou continues par morceaux, pÃĐriodiques. On reprend ici les notations du premier paragraphe âĒ The Fourier Series coefficients can be expressed in terms of magnitude and phase - Magnitude is independent of time (phase) shifts of x(t) - The magnitude squared of a given Fourier Series coefficient corresponds to the power present at the corresponding frequency âĒ The Fourier Transform was briefly introduce The Fourier cosine coefficient and sine coefficient are implemented in the Wolfram Language as FourierCosCoefficient[expr, t, n] and FourierSinCoefficient[expr, t, n], respectively. A Fourier series converges to the function (equal to the original function at points of continuity or to the average of the two limits at points of discontinuity b n = 4h nÏ when n is odd. So: f (x) = 4h Ï ( sin (x) + sin (3x) 3 + sin (5x) 5 + ) In conclusion: Think about each coefficient, sketch the functions and see if you can find a pattern, put it all together into the series formula at the end. And when you are done go over to: The Fourier Series Grapher

### Fourier coefficients Âŧ Chebfu

• The coefficient in the Fourier series expansion of is by default given by . The -dimensional Fourier coefficient is given by . In the form FourierCoefficient [ expr , t , n ] , n can be symbolic or an integer
• ed by its Fourier coefficients. The Fourier coefficients of an integrable function.
• Fourier Series Calculator is an online application on the Fourier series to calculate the Fourier coefficients of one real variable functions. Also can be done the graphical representation of the function and its Fourier series with the number of coefficients desired Input
• It is the discrete analog of the formula for the coefficients of a Fourier series : x n = 1 N â k = 0 N â 1 X k â e i 2 Ï k n / N , n â Z , {\displaystyle x_ {n}= {\frac {1} {N}}\sum _ {k=0}^ {N-1}X_ {k}\cdot e^ {i2\pi kn/N},\quad n\in \mathbb {Z} ,} (Eq.2) which is also. N {\displaystyle N} -periodic
• Fourier series calculator. Extended Keyboard; Upload; Examples; Random; Compute answers using Wolfram's breakthrough technology & knowledgebase, relied on by millions of students & professionals. For math, science, nutrition, history, geography, engineering, mathematics, linguistics, sports, finance, music Wolfram|Alpha brings expert-level knowledge and capabilities to the broadest possible.
• In this case, but not in general, we can easily find the Fourier Series coefficients by realizing that this function is just the sum of the square wave (with 50% duty cycle) and the sawtooth so. n: a n: b n: 0: 0.5----1: 0.6366: 0.6366: 2: 0-0.3183: 3-0.2122: 0.2122: 4: 0-0.1592: 5: 0.1273: 0.1273: 6: 0-0.1061: 7-0.0909 : 0.0909: From the relationship between the Trigonometric and Exponential.

The function returns the Fourier coefficients based on formula shown in the above image. The coefficients are returned as a python list: [a0/2,An,Bn]. a0/2 is the first Fourier coefficient and is a scalar. An and Bn are numpy 1d arrays of size n, which store the coefficients of cosine and sine terms respectively On the asymptotic behavior of the Fourier coefficients of $(1-R\cos\theta)^{-3/2}$ 1. Complex Fourier Series of $cos(x)$ 2. Why are the limits of the fourier cosine/sine series [0,â) while the limits of the fourier exponential series are (-â,â)? 1. Solving for coefficients of complex Fourier Series . Hot Network Questions Alien Instrument produces visual music, which a Human can. Fourier Coefficient. Fourier transformed infrared spectroscopy (FTIR) studies combined with CD spectroscopy confirmed the Îē-sheet formation between the peptides, while fluorescence emission spectra showed Ï-Ï stacking of the Fmoc groups. From: Self-assembling Biomaterials, 2018

### Fourier series - Wikipedi

3. e the Fourier Series coefficients C k as follows: 1. Take : 2. Multiply both sides of this equation by : 3. Integrate both sides of the equation over one period of the signal: Switching the order of the sum and integral in the second term (this is valid in all but pathological cases), we get . We've seen this integral before. We can pick any period over which to.
• In this section we define the Fourier Series, i.e. representing a function with a series in the form Sum( A_n cos(n pi x / L) ) from n=0 to n=infinity + Sum( B_n sin(n pi x / L) ) from n=1 to n=infinity. We will also work several examples finding the Fourier Series for a function
• so this could very well be an exciting video because we start with this idea of a Fourier series that we could take a periodic function and represent it as an infinite sum of weighted cosines and sines and we use that idea to say well can we find formulas for those coefficients and we were able to do that using the powers of calculus and now we can actually apply it for this particular square wave and I picked a square wave that has a period of 2pi and that's where actually a lot of these.
• ÃĐs, on peut maintenant donner la sÃĐrie de Fourier : Or b n = 0 pour tout n, et T = 2Ï donc Ï = 2Ï/T = 1, d'oÃđ : De plus, a n = 0 pour n pair (sauf a 0!!), donc on remplace n par 2k + 1 avec k âĨ 0 : Il n'y a plus qu'Ã  remplacer a 0 et a 2k + 1 avec l'expression trouvÃĐe prÃĐcÃĐdemment : On vient de dÃĐter

Look in the Results pane to see the model terms, the values of the coefficients, and the goodness-of-fit statistics. (Optional) Click Fit Options to specify coefficient starting values and constraint bounds, or change algorithm settings. The toolbox calculates optimized start points for Fourier series models, based on the current data set. You can override the start points and specify your own. Free Fourier Series calculator - Find the Fourier series of functions step-by-step This website uses cookies to ensure you get the best experience. By using this website, you agree to our Cookie Policy Fourier Series Coefficients via FFT (ÂĐ2004 by Tom Co) I. Preliminaries: 1. Fourier Series: For a given periodic function of period P, the Fourier series is an expansion with sinusoidal bases having periods, P/n, n=1, 2, p lus a constant. Given: f (t), such that f (t +P) =f (t) then, with P Ï=2Ï, we expand f (t) as a Fourier series by ( ) ( ) + + + = + + + b t b t f t a a t a t Ï Ï Ï.

### Fourier Series Coefficients Analysi

Computing Fourier series coefficients with the FFT. The Discrete Fourier Transform (DFT) is a mathematical function, and the Fast Fourier Transform (FFT) is an algorithm for computing that function. Since the DFT is almost always computed via the FFT, the distinction between the two is sometimes lost Solution attempt: Determining the fourier series of a function. (1) Theorem: For an even function over the symmetric range , the fourier series is given by. where. For we compute the Fourier coefficients. (2) The Fourier series of is. Apr 15, 2021. #6

### Finding Fourier coefficient

• Fourier's theorem states that any (reasonably well-behaved) function can be written in terms of trigonometric or exponential functions. We'll eventually prove this theorem in Section 3.8.3, but for now we'll accept it without proof, so that we don't get caught up in all the details right at the start. But what we will do is derive what the coe-cients of the sinusoidal functions must.
• Gold Member. 144. 33. It seems like it is asking you to find the Fourier coefficients of Ï ( x). Fourier coefficients are found by integrating the multiple of a function f ( x) and s i n or c o s ( 2 Ï r t T) with respect to time and where r is the order number of the coefficient. So r = 4 would be the fourth harmonic
• Fourier coefficients for all those segments are summed up to yield a Fourier coefficient histogram that is then used as the global feature vector. The actual classification is then conducted using the WEKA machine learning software suite . The microphone classification task is repeated with different classification algorithms and parameterizations for the feature extractor (non-overlapping.
• The adjusted form is 2 Ï Îī 0 â 1; this has Fourier coefficients. c n = { 1, n â  0 0, n = 0. Integration results in a function that jumps up by 2 Ï at 0, and is decreasing with slope â 1 the rest of the way. Such a function can be written as. f ( x) = { Ï â x, 0 < x < Ï â Ï â x, â Ï < x < 0. Now you can ignore the argument.

Als Fourierreihe, nach Joseph Fourier (1768-1830), bezeichnet man die Reihenentwicklung einer periodischen, abschnittsweise stetigen Funktion in eine Funktionenreihe aus Sinus- und Kosinusfunktionen.Die Basisfunktionen der Fourierreihe bilden ein bekanntes Beispiel fÃžr eine Orthonormalbasis.Im Rahmen der Theorie der HilbertrÃĪume werden auch Entwicklungen nach einem beliebigen. This article was adapted from an original article by T.P. Lukashenko (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098 The derivation of the Fourier series coefficients is not complete because, as part of our proof, we didn't consider the case when m=0. (Note: we didn't consider this case before because we used the argument that cos((m+n)Ï 0 t) has exactly (m+n) complete oscillations in the interval of integration, T). For the special case m=0  \begin{align The fourier transform is used to calculate the fourier coefficents ($\small c_n$) of a function. These coefficients are then used to express the function as weighted sum of harmonic sinusoids of different frequencies, phases and amplitudes. That's it

5 Responses to Fourier Coefficients Vincente E Zierk on October 16th, 2018 @ 12:07 am Hi is there any way to type in the numbers that i desire. hrm on October 27th, 2018 @ 11:49 am Dear Vincent, No. This is a design choice. Sliders offer a dynamic way to set variables in many of the Mathlets. If there are menu items that you think should be added, let's hear about them! J Sully on. 0. When doing a discrete fourier transform on some data using matlab's fft function, its output is a set of fourier coefficients but I was wondering how do I go about converting these into an and bn so I can reconstruct the signal using sines and cosines. E.g. I'd have a for loop that continually adds up a i cos (i x) + b isin (i x), where i = 1:N Fourier Series Calculator is a Fourier Series on line utility, simply enter your function if piecewise, introduces each of the parts and calculates the Fourier coefficients may also represent up to 20 coefficients. Derivative numerical and analytical calculato ### Fourierreihe - Wikipedi

Fourier Series of Even and Odd Functions. The Fourier series expansion of an even function $$f\left( x \right)$$ with the period of $$2\pi$$ does not involve the terms with sines and has the form: ${f\left( x \right) = \frac{{{a_0}}}{2} }+{ \sum\limits_{n = 1}^\infty {{a_n}\cos nx} ,}$ where the Fourier coefficients are given by the formulas \ 13. Suppose we are given the Fourier coefficients of an L 2 function on the circle. Are there necessary and sufficient conditions on the coefficients that allow us to determine that f is HÃķlder continuous of order Îą? Note that the necessary condition | f ^ ( n) | âĪ C f | n | â Îą is not sufficient. For example if f ^ ( n) = | n | â 2.

I am trying to compute the trigonometric fourier series coefficients of a periodic square wave time signal that has a value of 2 from time 0 to 3 and a value of -12 from time 3 to 6. It then repeats itself. I am trying to calculate in MATLAB the fourier series coefficients of this time signal and am having trouble on where to begin (b) Predict the convergence rate of the Fourier series coefficients, . (c) Find (directly) the exponential Fourier series for (í ĩíąĄ). (d) Compare the signal's exact power to that obtained using the dc and first 5 harmonic terms. (e) Plot the signal's spectra. (f) Verify your work employing the provided Mathcad exponential Fourier serie DLMF. Index; Notations; Search; Help? Citin FOURIER COEFFICIENTS + help The [Formula] key toggles display of the formula defining the Fourier coefficients a n and b n. The [Sine] and [Cosine] keys select between sine series or cosine series. Within each of these choices, the [All terms], [Odd terms], and [Even terms] keys restrict terms as stated. When possible, values of terms will be preserved under these changes, while the lost terms.

### SÃĐrie de Fourier â WikipÃĐdi

The Fourier transform allows us to deal with non-periodic functions. It can be derived in a rigorous fashion but here we will follow the time-honored approach of considering non-periodic functions as functions with a period T !1. Starting with the complex Fourier series, i.e. Eq. (14) and replacing X n by its de nition, i.e. Eq. (15), we obtain x(t) = X+1 n=1 1 T Z T=2 T=2 x(Ë)ei2Ënf 0 (t. In Section 40.3.2 we mentioned that the Fourier coefficients A n and B n can be calculated by fitting eq. (40.1) to the signal f(t) by a least squares regression.This fit is represented in a matrix notation as given in Fig. 40.36.The vector X represents the measurements, whereas A and B are vectors with respectively the real and imaginary Fourier coefficients Fourier Series--Square Wave. Consider a square wave of length .Over the range , this can be written a

Fourier analysis is a method for expressing a function as a sum of periodic components, and for recovering the signal from those components. When both the function and its Fourier transform are replaced with discretized counterparts, it is called the discrete Fourier transform (DFT). The DFT has become a mainstay of numerical computing in part because of a very fast algorithm for computing it. This is the implementation, which allows to calculate the real-valued coefficients of the Fourier series, or the complex valued coefficients, by passing an appropriate return_complex: def fourier_series_coeff_numpy (f, T, N, return_complex=False): Calculates the first 2*N+1 Fourier series coeff. of a periodic function Here is the simple online Fourier series calculator to do Fourier series calculations in simple. Just enter the values of f (x), upper & lower limit and number of coefficients, the calculator tool will fetch you the results automatically. In mathematics, a Fourier series is a method for representing a function as the sum of simple sine waves is called a Fourier series. Since this expression deals with convergence, we start by defining a similar expression when the sum is finite. Definition. A Fourier polynomial is an expression of the form which may rewritten as The constants a 0, a i and b i, , are called the coefficients of F n (x). The Fourier polynomials are -periodic functions. Using the trigonometric identities we can easily. Fourier series coefficients, it is typically preferable to think of the Fourier se-ries coefficients as a periodic sequence with period N, that is, the same period as the time sequence x(n). This periodicity is illustrated in this lecture through several examples. Partly in anticipation of the fact that we will want to follow an approach similar to that used in the continuous-time case for a.

I'm trying to compute the Fourier coefficients for a waveform using MATLAB. The coefficients can be computed using the following formulas: T is chosen to be 1 which gives omega = 2pi. However I'm having issues performing the integrals. The functions are are triangle wave (Which can be generated using sawtooth(t,0.5) if I'm not mistaking) as well as a square wave. I've tried with the following. Calculate the Fourier coefficients of the series expansion of a function, and the amplitude and phase spectra. The script contains some theory and 3 different methods to calculate the coefficients. USAGE fourier_coeff(fun,t0,T) fourier_coeff(fun,t0,T,M) fourier_coeff(fun,t0,T,M,N) fourier_coeff(fun,t0,T,M,N,method) fourier_coeff(fun,t0,T,M,N,method,res) fourier_coeff(fun,t0,T,M,N,method,res.

### Fourier Series -- from Wolfram MathWorl

pairs, yielding a coefficient vector -Fast Fourier Transform (FFT) can perform DFT and inverse DFT in time Î(í ĩíąlogí ĩíą) âĒAlgorithm 1. Add í ĩíą higher-order zero coefficients to ( ) and ( ) 2. Evaluate ( ) and ( ) using FFT for 2í ĩíą points 3. Pointwise multiplication of point-value forms 4. Interpolate ( ) using FFT to compute inverse DFT 18 . Complex Roots of Unity âĒA complex í ĩíēth. Hello I want to find the Fourier series and/or the coefficients for a function like the following: or or . For the first one I did the following: FourierTrigSeries[ Piecewise[{{0, -Pi <= x <= -Pi/2}, {Cos[x], -Pi/2 <= x <= Pi/2}, {0, Pi/2 < x < Pi}}], x, 5 ] Which seems to be correct. How can I convert this to summation form? For Taylor Series I'm using something like this: series[expr_, x_,

### Fourier Series - mathsisfun

Example sentences with Fourier coefficient, translation memory. add example. en The result of such a procedure are the Fourier coefficients Ao, An (cosine coeff.) and Bn(sine coeff.). springer. de FÃžr die Analyse wurden N=48 Werte benutzt. en Emphasis is laid on the rigorous analysis of line profiles in terms of Fourier coefficients. springer. de Besonders betont wird die Analyse der. Many translated example sentences containing Fourier coefficients - German-English dictionary and search engine for German translations Recognize that each Fourier component corresponds to a sinusoidal wave with a different wavelength or period. Mentally map simple functions between Fourier space and real space. Describe sounds in terms of sinusoidal waves. Describe the difference between waves in space and waves in time. Recognize that wavelength and period do not correspond to specific points on the graph but indicate the.

### FourierCoefficientâWolfram Language Documentatio

Introduction to Complex Fourier Series Nathan P ueger 1 December 2014 Fourier series come in two avors. What we have studied so far are called real Fourier series: these decompose a given periodic function into terms of the form sin(nx) and cos(nx). This document describes an alternative, where a function is instead decomposed into terms of the form einx. These series are called complex. In other words, Fourier coefficients of frequency-distance 0 from the origin will be multiplied by 0.5. As you go away from the origin or zero frequency, out to frequency-distance 96, the multiplier will be interploated between 0.5 and 4.0. From then outward, the multiplier will be 4.0. So higher frequency coefficients are multiplied by values greater than 1.0 and lower frequency coefficients. Fourier Transform Coefficients Of Real Valued Audio Signals 2018-02-10 - By Robert Elder. This article is effectively an appendix to the article The Fast Meme Transform: Convert Audio Into Linux Commands.In this article, we will review various properties of the coefficients that result from applying the Discrete Fourier transform to a purely real signal

### Fourier Coefficient Article about Fourier Coefficient by

Fourier series coefficients. Consider an audio signal given by s ( t) = sin (440Ã 2Ï t) + sin (550 Ã 2Ï t) + sin (660Ã 2Ï t ). This is a major triad in a non-well-tempered scale. The first tone is A-440. The third is approximately E, with a frequency 3/2 that of A-440. The middle term is approximately C sharp, with a frequency 5/4 that of. The Fourier series, Fourier transforms and Fourier's Law are named in his honour. Jean Baptiste Joseph Fourier (21 March 1768 - 16 May 1830) Fourier series. To represent any periodic signal x(t), Fourier developed an expression called Fourier series. This is in terms of an infinite sum of sines and cosines or exponentials. Fourier series uses. Fourier coefficients. âĶ 31â35 of 35 matching pages âĶ . Search Advanced Help (0.001 seconds) 31â35 of 35 matching pages 31: 29.6 Fourier Series Â§29.6 Fourier Series with Îą p, Îē p, and Îģ p as in (29.3.11) and (29.3.12), and An alternative version of the Fourier series expansion (29.6.1) is given by with Îą p, Îē p, and Îģ p as in (29.3.13) and (29.3.14), and 29.6.20 â p. On the Parseval's Formula for the Sum of Coefficients of a Fourier Series page, for \$\displaystyle{s_n(x) = \sum_{k=0}^{n} c_k\varphi_k(x)}. The real and imaginary parts of the Fourier components of a square wave (assumed periodic with a period of 256) as a function of the square wave width and position are shown in the graph on the right. The Fourier components are normalized to lie within or on the unit circle (shown in red). The vertical axis is the imaginary part and the horizontal axis is the real part respectively

### Fourier Series Calculator - Fourier Series on line

To find the Fourier coefficient of , we take complex conjugation on both sides of the Fourier expansion of : and replace by . i.e., In particular, if is real, we have: In other words, if is real, the real and imaginary parts of its Fourier coefficient are even and odd, respectively. Parseval's Relation . In the equation for the theorem of frequency convolution if we assume (a) and (b) , and. Azimuthal Fourier Coefficients provide a compact means to describe the azimuthal reflectivity of prestack seismic data. The FCs are primarily descriptive in nature, with each coefficient providing independent information. The parameterization is parsimonious with only a few nonzero coefficients. For PP data, the odd coefficients are all zero. The linearized PP reflectivity for arbitrary. To illustrate determining the Fourier Coefficients, let's look at a simple example. The cosine function, f(t), is shown in Figure 1: Figure 1. The Cosine Function. Now, it may be obvious to some what the Fourier Coefficients are, but it is still worth finding the coefficients to ensure the process is understood. This function is mathematically described by the equation:  First, we need to. Fourier coefficient allows the function to be represented in the form of the trigonometric function. Answer and Explanation: 1. Become a Study.com member to unlock this answer! Create your account. ### Discrete Fourier transform - Wikipedi

This Demonstration illustrates recovering the Fourier coefficients from a complex wave that you build. With the sliders you can select the weights of five sine wave signals, 1 to 5 Hz. These are summed into a complex signal in the upper graph. You can then selectively choose to multiply the entire output wave by any of the original unweighted signals âĒ Using the procedure to measure the Fourier coefficients it is possible to predict the amplitude of each harmonic tone. Predicting the spectrum of a plucked string âĒYou know the shape just before it is plucked. âĒYou know that each mode moves at its own frequency âĒThe shape when released âĒWe rewrite this as . Predicting the motion of a plucked string (continued) Each harmonic has its. Fourier Series, Integrals, and, Sampling From Basic Complex Analysis JeïŽrey RAUCH Outline. The Fourier series representation of analytic functions is derived from Laurent expan-sions. Elementary complex analysis is used to derive additional fundamental results in harmonic analysis including the representation of Câ periodic functions by Fourier series, the representation of rapidly.  I have managed to find the rms value of the output voltage, but haven't a clue where to start with finding the Fourier coefficients requested in the question below: A single-phase ac voltage regulator has a resistive load of R= 10 ohms, and the input voltage is Vs = 240 V, 50 Hz. The delay.. The figure above shows a set of periodic signals (left) and their Fourier expansion coefficients (right) as a function of frequency (real and imaginary parts are shown in solid and dashed lines, respectively). The first three rows show two sinusoids and , and their weighted sum . The following four rows are for the impulse train, square wave, triangle wave, and sawtooth wave, respectively. As of now, you should have a better understanding of the Fourier coefficients and the different types of symmetry that can happen. These five types, even, odd, half-wave, quarter-wave half-wave even, and quarter-wave half-wave odd are all used to simplify the computation of the Fourier coefficients. A few topics that will be covered next will go in depth to find the steady-state response of a. Finding the coefficients, F' m, in a Fourier Sine Series Fourier Sine Series: To find F m, multiply each side by sin(m't), where m' is another integer, and integrate: But: So: Ãonly the m' = m term contributes Dropping the ' from the m: Ãyields the coefficients for any f(t)! f (t) = 1 Ï F mâē sin(mt) m=0 ââ Kohnen, W.: Fourier coefficients of modular forms of half-integral weight. Math. Ann.271, 237-268 (1985) Google Scholar 9. Mordell, L.J.: The sign of the Gaussian sum. Ill. J. Math.6, 177-180 (1962) Google Scholar 10. Niwa, S.: Modular forms of half-integral weight and the integral of certain theta-functions

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