103年 - 103 國立臺北大學_學士班曁進修學士班轉學生招生考試試題_電機工程學系、通訊工程學系學士班3年級：工程數學#147398

1. (10%) Let $a_n = i/2^{3n}$ and $b_n = 1/2^{3n+1}$. Is the following series convergent or divergent?
$a_0 + b_0 + a_1 + b_1 + \cdots = i + \frac{1}{2} + \frac{i}{8} + \frac{1}{16} + \frac{i}{64} + \frac{1}{128} + \cdots$

2. (10%) Find all Taylor and Laurent series with center 0 of
$f(z) = \frac{-2z + 3}{z^2 - 3z + 2}.$

3. (10%) Integrate the following function counterclockwise around C. Show the details.
$\frac{15z + 9}{z^3 - 9z}, C: |z| = 4$

4. (10%) Use complex integration method to find
$\int_{-\infty}^{\infty} \frac{dx}{x^2 - 4ix}$

5. (10%) Prove the following Fourier transform operation.
$v(t - \tau) \xrightarrow{F} e^{-j2\pi f\tau} \overline{V}(f)_v$

6. (8%) Solve
$[y^2 - e^x \sin(e^x + 1)]dx + 2xydy = 0.$

7. (8%) Solve
$(x + 2y + 3)dx + (x - 2y + 4)dy = 0.$

8. (8%) Find the orthogonal trajectories of $(x - c)^2 + y^2 = c^2$.

9. (8%) Solve
$\frac{d^2y}{dx^2} + y = x + \sec x.$

10. (8%) Find the Fourier series of the period function $f(x) = x + \pi, |x| < \pi$.

11. (10%) Solve the following initial value problem by the Laplace transform.
$\begin{cases} y_1'(x) = y_1(x) - 2y_2(x) + 1, \\ y_2''(x) = -3y_1(x) - 2y_2(x), \\ y_1(0) = y_2(0) = y_2'(0) = 0. \end{cases}$