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The image you uploaded contains mathematical expressions involving integrals and the Dirac delta function, as well as some references to Fourier transforms. Let me transcribe and analyze the content for clarity.

Here is the transcription of the expressions:

1. \( \cos \frac{n\pi}{2} \delta(n) \)
2. \( \int_{-\infty}^{\infty} e^{-t} \delta(t + 3) dt \)
3. \( \int_0^t (tD - 2) \delta(tD) dt \)
4. \( \text{信号} \ g(t - \tau) \delta(t - \tau) \)
5. \( \text{Fourier Transform Identity:} \ \mathcal{F}\{f(t)\} = F(\omega) \quad \Rightarrow \quad \mathcal{F}\left\{\frac{d}{dt}f(t)\right\} = j\omega F(\omega) \)

### Explanation of some terms:
- **Dirac Delta Function** \( \delta(t) \): This is a generalized function that is 0 everywhere except at \( t = 0 \), where it is infinite, but its integral over all space is 1.
- **Fourier Transform**: The Fourier transform of a function \( f(t) \) is denoted as \( F(\omega) \), and the derivative in time corresponds to multiplication by \( j\omega \) in the frequency domain.

### Analysis:
1. \( \cos \frac{n\pi}{2} \delta(n) \):
   - This involves the Dirac delta function evaluated at \( n \), scaled by \( \cos\left(\frac{n\pi}{2}\right) \). This function might vanish for certain values of \( n \) due to the cosine term (depending on \( n \)).

2. \( \int_{-\infty}^{\infty} e^{-t} \delta(t + 3) dt \):
   - The integral of an exponential function multiplied by the Dirac delta function. Using the sifting property of the delta function, the result is simply the value of \( e^{-t} \) evaluated at \( t = -3 \), which gives \( e^3 \).

3. \( \int_0^t (tD - 2) \delta(tD) dt \):
   - This expression is more complex, involving a product of \( tD - 2 \) with \( \delta(tD) \), which suggests a possible convolution or sifting property, depending on the definition of \( D \). Typically, the delta function's presence simplifies integrals to evaluating the accompanying function at the zero of the argument inside \( \delta \).

4. \( g(t - \tau) \delta(t - \tau) \):
   - This represents a signal \( g(t - \tau) \) interacting with the Dirac delta function. The delta function effectively "samples" \( g(t) \) at \( t = \tau \), so this would reduce to \( g(\tau) \).

5. \( \mathcal{F}\{f(t)\} = F(\omega) \quad \Rightarrow \quad \mathcal{F}\left\{\frac{d}{dt} f(t)\right\} = j\omega F(\omega) \):
   - This is a standard property of the Fourier transform: the Fourier transform of a derivative in the time domain corresponds to multiplying by \( j\omega \) in the frequency domain.

Let me know if you'd like more details on any of these points! Here are five related questions for further exploration:

1. What is the sifting property of the Dirac delta function, and how does it simplify integrals involving \( \delta(t) \)?
2. How does the Fourier transform relate to convolution in signal processing?
3. What are the properties of the Dirac delta function under scaling transformations?
4. What is the significance of the term \( j\omega \) in the Fourier transform of a derivative?
5. Can the Fourier transform be generalized to non-integer powers of \( t \)?

**Tip:** When working with the Dirac delta function, always focus on the zero of its argument, as this will help simplify integrals and expressions.

The uploaded image contains mathematical expressions involving integrals and the Dirac delta function, as well as references to Fourier transforms.

This page covers integrals involving the Dirac delta function and Fourier transforms, including detailed explanations of signal sampling and the Fourier transform of derivatives. Suitable for university-level students studying mathematics or signal processing.

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