📘 mathematical analysis

1. **State the problem:** Find the Fourier sine transform of the function $$f(x) = x$$ defined on the interval $$0 < x < a$$. 2. **Recall the formula for the Fourier sine transform

1. **Problem statement:** Find the Fourier cosine transform of the piecewise function $$f(x) = \begin{cases} x, & 0 < x < \frac{1}{2} \\ 1 - x, & \frac{1}{2} < x < 1 \\ 0, & x > 1

1. The problem is to understand the definition of a function $f(t)$ being of exponential order $e^{\alpha t}$.\n\n2. A function $f(t)$ is said to be of exponential order $e^{\alpha

1. **Problem Statement:** Find the Fourier Transform of the function $$f(x) = \begin{cases} 1 & \text{if } |x| < 1 \\ 0 & \text{if } |x| > 1 \end{cases}$$ and then use it to evalua

1. المشكلة: نريد فهم الفرق بين الحد العلوي (supremum) والحد الأعلى (infimum) في سياق المتتاليات أو المجموعات العددية. 2. التعريفات:

1. **Prove: If** $(\sum u_n)$ **converges, then** $(\sum \frac{\sqrt{u_n}}{n})$ **converges.** Since $(\sum u_n)$ converges with $u_n \geq 0$, by Cauchy-Schwarz inequality: