Fourier Transform Of E Ax

Decoding the Fourier Transform of eax: A Question-and-Answer Approach

The Fourier transform is a powerful mathematical tool that decomposes a function into its constituent frequencies. It's ubiquitous in fields ranging from signal processing and image analysis to quantum mechanics and financial modeling. Understanding the Fourier transform of simple functions forms a crucial foundation for applying this technique to more complex scenarios. This article explores the Fourier transform of the exponential function, eax, providing a detailed, question-and-answer approach.

I. What is the Fourier Transform, and Why is eax Important?

Q: What is the Fourier Transform?

A: The Fourier transform essentially converts a function from the time domain (or spatial domain) to the frequency domain. Instead of describing a signal by its amplitude at different points in time, it describes it by its amplitude at different frequencies. Imagine a musical chord: the time domain representation would be the waveform over time, while the frequency domain representation would be the individual notes that make up the chord.

Q: Why is the Fourier transform of eax significant?

A: The exponential function eax is a fundamental building block in many mathematical models. Its Fourier transform provides a crucial stepping stone for understanding the transforms of more complex functions. Many signals and systems can be represented (or approximated) as combinations of exponentials, making this transform essential for analysis.

II. Deriving the Fourier Transform of eax

Q: What is the definition of the Fourier Transform?

A: The continuous Fourier Transform of a function f(t) is defined as:

F(ω) = ∫-∞∞ f(t)e-jωt dt

where:

F(ω) is the Fourier transform of f(t)
ω is the angular frequency (radians per second)
j is the imaginary unit (√-1)
the integral is taken over all time.

Q: How do we derive the Fourier Transform of f(t) = eat?

A: We substitute f(t) = eat into the Fourier transform definition:

F(ω) = ∫-∞∞ eate-jωt dt = ∫-∞∞ e(a-jω)t dt

This integral evaluates to:

F(ω) = [e(a-jω)t / (a-jω)] -∞∞

This integral converges only if the real part of (a-jω) is negative, meaning 'a' must be negative. If 'a' is negative, the result is:

F(ω) = 2πδ(ω) where δ is the Dirac delta function. If 'a' is negative, we have:

F(ω) = 2πδ(ω-ja)

Q: What does this result mean?

A: The Dirac delta function, δ(x), is zero everywhere except at x=0, where it is infinitely high. This means the Fourier transform of eat (for a < 0) is zero everywhere except at the imaginary frequency ω = ja. This demonstrates that a decaying exponential in the time domain corresponds to a single point in the frequency domain.

III. Real-World Applications

Q: Where is this transform applied in real-world scenarios?

A: The Fourier transform of eat, and its more general form involving complex exponentials, is critical in:

System analysis: Many linear time-invariant systems have impulse responses that can be modeled as decaying exponentials. The transform helps analyze the system's frequency response.
Signal processing: Exponential decay represents the behavior of many signals (e.g., the decay of a voltage in an RC circuit). The Fourier transform helps in isolating different frequencies within the signal for filtering or analysis.
Nuclear Magnetic Resonance (NMR): The signal decay in NMR experiments is often exponential, and its Fourier transform gives the spectrum of frequencies associated with different nuclei, providing crucial information for chemical analysis.

IV. Takeaway

The Fourier transform of eat, while seemingly simple, highlights the core concept of frequency decomposition. It shows the relationship between an exponential decay in the time domain and a specific frequency (or, more precisely, an impulse at an imaginary frequency) in the frequency domain. This forms the foundation for understanding the transforms of much more complex functions and plays a vital role in numerous scientific and engineering disciplines.

V. FAQs:

1. What happens if 'a' is positive or complex? If 'a' is positive, the integral diverges, indicating the function isn't Fourier transformable in the traditional sense. For complex 'a', the transform exists but involves a more complex expression involving Dirac delta functions.

2. How is this related to the Laplace Transform? The Laplace transform is a generalization of the Fourier transform, allowing for the analysis of functions that aren't Fourier transformable (like those with 'a' positive). The Fourier transform can be considered a special case of the Laplace transform.

3. Can this transform be used with discrete signals? Yes, the Discrete-Time Fourier Transform (DTFT) and the Discrete Fourier Transform (DFT) are the discrete counterparts, suitable for analyzing digital signals.

4. What is the inverse Fourier transform of the result? The inverse Fourier transform of 2πδ(ω-ja) is indeed eat (for a<0), demonstrating the reversibility of the transform.

5. How can I compute this transform numerically? Software packages like MATLAB, Python (with libraries like NumPy and SciPy), and others provide functions for numerical computation of the Fourier transform, allowing you to approximate the transform even for functions that lack closed-form solutions.

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How to calculate Fourier Transform of e^-a*|t|? - Physics Forums 2 Apr 2014 · Calculate (from the definition, no tables allowed) the Fourier Transform of [itex]e^{-a*|t|}[/itex], where a > 0. Homework Equations Fourier Transform: [itex]G(f) = \int_{-\infty}^{\infty} g(t)e^{-j\omega t} dt[/itex] The Attempt at a Solution I thought I'd break up the problem into the two cases of t (where it's negative and positive).

calculus - What is the Fourier transform of $f(x)=e^{-x^2 ... 14 Jun 2018 · Caveat: I'm using the normalization $\hat f(\omega) = \int_{-\infty}^\infty f(t)e^{-it\omega}\,dt$. A cute way to to derive the Fourier transform of $f(t) = e^{-t^2}$ is the following trick: Since $$f'(t) = -2te^{-t^2} = -2tf(t),$$ taking the Fourier transfom of both sides will give us $$i\omega \hat f(\omega) = -2i\hat f'(\omega).$$

1 Fourier Transform - University of Toronto Department of … We introduce the concept of Fourier transforms. This extends the Fourier method for nite intervals to in nite domains. In this section, we will derive the Fourier transform and its basic properties. L gn2Z. Consider the inner product for complex valued functions. so the complex exponentials are an orthogonal set.

integration - Fourier transform of $e^{-ax^2}$ in $\mathbb{R}^2 ... 23 Jan 2019 · The Fourier transform of $f_1(x) = e^{-ax^2}$ is $\sqrt{\frac{\pi}{a}}e^{-\frac{\xi^2}{4a}}$ and the Fourier transform of $f_2(y) = 1$ is $2\pi \delta(\eta)$. So, we obtain the formula $$\mathcal F(f) =2\pi \sqrt{\frac{\pi}{a}}e^{-\frac{\xi^2}{4a}}\delta(\eta).$$

Fourier Transform--Exponential Function - Wolfram MathWorld 31 Jan 2025 · The Fourier transform of e^(-k_0|x|) is given by F_x[e^(-k_0|x|)](k)=int_(-infty)^inftye^(-k_0|x|)e^(-2piikx)dx = int_(-infty)^0e^(-2piikx)e^(2pixk_0)dx+int_0^inftye^(-2piikx)e^(-2pik_0x)dx (1) = (2) Now let u=-x so du=-dx, then F_x[e^(-k_0|x|)](k) =int_0^infty[cos(2piku)+isin(2piku)]e^(-2pik_0u)du +int_0^infty[cos(2piku)-isin(2piku)]e^(-2pik ...

Fourier Transformation of $e^{-a|x|}$ - Mathematics Stack Exchange Split the integral into two regions and use the fact that $\vert x\vert = -x$ for $x \lt 0$ and $\vert x\vert = x$ for $x\gt 0$: $$ F_\text{trans}= \int_{-\infty} ^{\infty} f(x) e^{-ikx}\,dx = \int_{-\infty} ^{\infty} e^{-a|x|} e^{-ikx}\,dx \\ = \int_{-\infty} ^{0} e^{ax} e^{-ikx}\,dx + \int_{0} ^{\infty} e^{-ax} e^{-ikx}\,dx \\ = \int_{-\infty ...

E: Fourier transforms - Wiley Online Library Fourier transforms occur naturally and ubiquitously in the mathematical description of scattering. The reason is that the scattering amplitude from an extended body often appears as a Fourier transform. Here we remind the reader of a few important definitions, and work through some illustrative examples. 2π .

calculus - what is the Fourier cosine transform of $e^{-ax ... 28 Apr 2015 · What is the Fourier cosine transform of $e^{-ax}$ I got $$ \int_{0}^{\infty}\cos(kx)e^{-ax}dx = \frac{e^{-ax}(k\sin(kx) -\cos(kx))} {a^{2}+k^{2}}\Bigr|_{0}^{\infty} $$ But how do you continue from here?

The Fourier transform - The Department of Mathematics The Fourier transform For a function f(x) : [ L;L] !C, we have the orthogonal expansion f(x) = X1 n=1 c ne inˇx=L; c n = 1 2L Z L L f(y)e inˇy=Ldy: Formal limit as L !1: set k n = nˇ=L and k = ˇ=L f(x) = 1 2ˇ X1 n=1 Z L L f(y)e iknydy! eiknx k: This is a Riemann sum: k !0 gives f(x) = 1 2ˇ Z 1 1 F(k)eikxdk; where F(k) = Z 1 1 f(x)e ikxdx ...

Fourier Transform -- from Wolfram MathWorld 14 Feb 2025 · The Fourier transform is a generalization of the complex Fourier series in the limit as L->infty. Replace the discrete A_n with the continuous F(k)dk while letting n/L->k. Then change the sum to an integral, and the equations become f(x) = int_(-infty)^inftyF(k)e^(2piikx)dk (1) F(k) = int_(-infty)^inftyf(x)e^(-2piikx)dx.

Appendix A Fourier transforms - Heidelberg University • Winding in x leads to a shift in u: The Fourier transform of e2πiu0xf(x) is g(u−u 0). • The Fourier transform of f(ax) where a is a non-zero real number is g(u/a)/|a|. • The Fourier transform of f∗(x) (the complex conjugate) is g∗(−u). • If f(x) is real, then g(−u)=g∗(u) (i.e. the Fourier transform of …

real analysis - Fourier Series Representation $e^{ax} 7 Mar 2015 · a) Compute the full Fourier series representation of $f(x) = e^{ax}, −π ≤ x < π.$ b) By using the result of a) or otherwise determine the full Fourier series expansion for the function $g(x)=\

2 Fourier transforms - University of Bristol We will use Fourier Cosine and Sine transforms for solving pde problem in semi-infinite domains. but this can not be evaluated using elementary techniques. Instead we treat integral in complex. k-plane, where the integrand has simple poles at k = ±i.

Fourier Transform of Generalized Functions | SpringerLink 9 Feb 2025 · Unfortunately, the integral equation does not satisfy the integrability criterion (Eq.4.5).Using the principle of duality, we have established the inverse Fourier transform of $sinc \left (x\right )$ as $\Pi \left (x\right )$.The method is quite restrictive and not useful for all classes of functions we encounter in physics or engineering.

Fourier Sine & Cosine Transform || e^(-ax) || 18mat31 - YouTube In this video, we have calculated the Fourier Sine and Cosine transform of e^ (-ax) in easy steps and detailed formula.#DrPrashantPatil#FourierTransforms#18MA...

Fourier transform - Wikipedia An example application of the Fourier transform is determining the constituent pitches in a musical waveform.This image is the result of applying a constant-Q transform (a Fourier-related transform) to the waveform of a C major piano chord.The first three peaks on the left correspond to the frequencies of the fundamental frequency of the chord (C, E, G).

4 Fourier transform - Texas A&M University Solution: Using (a) we deduce that g(!) = F(f)(!), that is to say, F 1(g)(x) = F 1(F(f))(x). Now, using the inverse Fourier transform, we deduce that F 1(g)(x) = f(x) at every point x where f(x) is of class C1 and F 1(g)(x) = 2 (f(x ) + f(x+)) at discontinuity points of f. As a result:

Table of Fourier Transform Pairs - ETH Zürich Shows that the Gaussian function exp( - at2) is its own Fourier transform. For this to be integrable we must have Re(a) > 0. it's the generalization of the previous transform; Tn (t) is the Chebyshev polynomial of the first kind.

Lecture 16: Fourier transform - MIT OpenCourseWare Compare Fourier and Laplace transforms of x(t) = e −t u(t). a complex-valued function of complex domain. a complex-valued function of real domain. The Laplace transform maps a function of time t to a complex-valued function of complex-valued domain s.

8 Fourier Transforms - University of Cambridge For a (non-periodic) function f : R C, we likewise define the Fourier transform ̃f(k) of f(x) to be30. ̃f(k) := e ikx f(x) dx . The Fourier transform is an example of a linear transform, producing an output function ̃f(k) from the input f(x).

Fourier transform - Purdue University e ax2 =2e isxdx= C(a)e s2 (2a) and use the result that Z 1 1 e ax2=2dx= r 2ˇ a; proved in the handout \Some de nite integrals". Putting a= 1 in (5) we obtain that f(x) = e x2 =2has Fourier transform p 2ˇe s2; (6) in other words, it is an eigenfunction of the Fourier operator F with eigen-value p 2ˇ. Formula (6) is very important, and it is ...

Computing the Fourier transform of $e^ {-|x|}$ Wolfram Alpha defines the Fourier transform of an integrable function as $$ \hat {f} (\xi ) =\frac {1} {\sqrt {2\pi}} \int_ {-\infty}^ {\infty} f (x) e^ {-i\xi x} \, dx, $$ while the inverse Fourier transform is taken to be $$ \check {f} (\xi ) =\frac {1} {\sqrt {2\pi}} \int_ {-\infty}^ {\infty} f (x) e^ {i\xi x} \, dx. $$ If you check your solu...

Fourier Transform Of E Ax

Decoding the Fourier Transform of e<sup>ax</sup>: A Question-and-Answer Approach

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