Transform of $\frac{u^3}{u^4+4}$: Fourier Analysis

Thread starter nhrock3
Start date Dec 30, 2009
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Fourier Fourier transform Transform

In summary, the Fourier transform of $\frac{u^3}{u^4+4}$ is $\hat{f}(\omega) = \int_{-\infty}^{\infty} \frac{u^3}{u^4+4}e^{-i\omega t} dt$, which can be calculated using standard techniques of Fourier analysis and has a domain of all possible values of $\omega$ and a range of complex numbers. It can also be used to solve differential equations and there are other transforms that can be used for analysis.

Dec 30, 2009

nhrock3

[tex]\frac{u^3}{u^4+4}[/tex]
what is the transform of this function?

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Dec 30, 2009

White

if you ask just for the result here it is:
http://www.wolframalpha.com/input/?i=Fourier+transform+u^3%2F%28u^4%2B4%29

Related to Transform of $\frac{u^3}{u^4+4}$: Fourier Analysis

1. What is the Fourier transform of $\frac{u^3}{u^4+4}$?

The Fourier transform of a function $f(t)$ is defined as $\hat{f}(\omega)=\int_{-\infty}^{\infty} f(t)e^{-i\omega t} dt$. In this case, the Fourier transform of $\frac{u^3}{u^4+4}$ is $\hat{f}(\omega) = \int_{-\infty}^{\infty} \frac{u^3}{u^4+4}e^{-i\omega t} dt$.

2. How do you calculate the Fourier transform of $\frac{u^3}{u^4+4}$?

The Fourier transform of $\frac{u^3}{u^4+4}$ can be calculated using the standard techniques of Fourier analysis, such as integration by parts and substitution. The resulting integral will depend on the value of $\omega$, and may require the use of partial fractions to simplify.

3. What is the domain and range of the Fourier transform of $\frac{u^3}{u^4+4}$?

The domain of the Fourier transform is the set of all possible values of $\omega$, which is the frequency domain. The range of the Fourier transform is the set of complex numbers, as the transform itself is a complex-valued function.

4. Can the Fourier transform of $\frac{u^3}{u^4+4}$ be used to solve differential equations?

Yes, the Fourier transform can be used to solve differential equations that have initial or boundary conditions. By taking the Fourier transform of both sides of the equation, the resulting algebraic equation can be solved for the Fourier transform of the solution, which can then be transformed back to the time domain to obtain the solution.

5. Are there any other transforms that can be used to analyze $\frac{u^3}{u^4+4}$?

Yes, there are other transforms such as the Laplace transform and the Z-transform that can also be used to analyze $\frac{u^3}{u^4+4}$. Each transform has its own advantages and applications, so it is important to choose the appropriate transform based on the problem at hand.