Question

算个广义积分

被积函数： f(x)=sin(z x)sin(w x)/x
积分区间：(0,+无穷)

Answer 1

不妨设z > w > 0 (z = ±w时发散, 而负号容易提出), 进一步可设z = aw, a>1,
则∫{0,+∞} sin(zx)sin(wx)/xdx = ∫{0,+∞} sin(awx)sin(wx)/x dx = ∫{0,+∞} sin(ax)sin(x)/x dx.
定义含参变量y的广义积分f(y) = ∫{0,+∞} e^(-xy)sin(ax)sin(x)/x dx.
交换积分与求导(应该可以验证满足某交换条件), f'(y) = -∫{0,+∞} e^(-xy)sin(ax)sin(x)dx.
这个原函数是初等函数, 积分可算得-1/2·y/((a-1)²+y²)+1/2·y/((a+1)²+y²).
于是f(y) = C-1/4·ln((a-1)²+y²)+1/4·ln((a+1)²+y²).
注意到|f(y)| ≤ ∫{0,+∞} e^(-xy)dx = 1/y → 0当y → +∞, 由此确定C = 0.
∫{0,+∞} sin(zx)sin(wx)/xdx = f(0) = -1/2·ln((a-1)/(a+1)) = -1/2·ln((z-w)/(z+w)).
讨论符号之后得对任意z ≠ ±w, ∫{0,+∞} sin(zx)sin(wx)/xdx = -1/2·ln|(z-w)/(z+w)|.

Answer 2

-1/2 * ln|(w-z)/(w+z)|

追问

过程呢？