1 First, prove the formula, which can be proved by mathematical induction: that is, just prove L (t * f (t)) = df (s)/ds; That is, N= 1 holds, and f (s)' = integral (d (f (t) * e (-st))/ds) dt =-integral (t * f (t) * e (-st)) dt =-l [t * f (t). Obviously, mathematical induction can get the conclusion you want:

Direct differential integral transformation table can do your topic: 6s/(s 2+9) 2. The differential Laplace integral transformation table can be obtained as follows: l [Tsin (at)] = 2as/(S2+A2) 2; Obviously, a=3 can be obtained by comparing the tables, and the answer is Tsin (3t);

2. First of all, Fourier coefficients are not obtained by integral transformation, but by the orthogonality of trigonometric functions. Fourier transform and Fourier series expansion are different, one is for arbitrary function and the other is for periodic function, so don't confuse them. The method of solving Fourier series coefficients is as follows

A0= 1/L* integral (f(t)dt, -l, L)an= 1/L* integral (f (t) cos (n * pi * t/l) dt, -L, l), BN = 65438+. ....

Laplace transform is developed on the basis of Fourier transform, which has many good properties that Fourier does not have, but they are very different and their application scope is different. Integral transformation and expansion of function coefficients into series are two different concepts. Don't confuse them. I don't think it necessary to answer the second question. If you have any questions, please call me 1.

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Exponential representation of Fourier series, also known as complex form of Fourier series:

Euler's formula shows that e IX = cos (x)+isin (x). Through this formula, we can express the complex exponential expressions of sin (t) and cos (t), transform the original expression of Fourier trigonometric series into the negative exponential expression of Fourier, and establish the corresponding relationship between their coefficients as follows:

We know: f (x) = A0/2+sum (an * cos (n * pi * x/l)+bn * sin (n * pi * x/l));

For the above expressions, the complex expressions of SIN (t) and COS (t) expressed by Leo formula are:

f(x)=a0/2+sum(an/2*(e^(i*n*pi*x/l)+e^(-i*n*pi*x/l))-i*bn/2*(e^(i*n*pi*x/l)-e^(-i*n*pi*x/l)))=a0/2+sum((an-i*bn)/2*e^(i*pi*n*x/l))+sum((an+i*bn)/2*e^(-i*pi*n*x/l))；

Let cn = (an-I * bn)/2; c-n =(an+I * bn)/2； c0 = A0/2；

Then the original sequence is expressed as: f (x) = sum (cn * e (I * n * pi * x/l));