110年 - 110 國立臺灣科技大學_碩士班招生試題_自動化及控制研究所：工程數學#101292

(1)(10%)

(2) (10%)

4. Determine the Fourier cosine transform and Fourier sine transform of the following function., if K is any positive number. (10%) 

5. Solve the telegraph equation

 for 0 ＜x ＜L, t ＞ 0. A and B are positive constants.
The boundary conditions are
u(0,t)=u(L,t)=0 for t ≥ 0,
and the initial conditions are

 Assume that A²L² ＜4(BL²+c²π²) (20%)

6. Use convolution to find the inverse Fourier transform of the function.
(10%)

7. Evaluate

with γ as the circle |z| = 2 first by using the residue theorem and then by using the argument principle. (10%)