∫
−
∞
+
∞
e
−
x
2
sin
2
(
x
2
)
x
2
d
x
\int_{-\infty}^{+\infty}\frac{e^{-x^{2}}\sin^{2}\left(x^{2}\right)}{x^{2}}\mathrm{d}x
∫−∞+∞x2e−x2sin2(x2)dx
解:
令:
I
=
∫
−
∞
+
∞
e
−
x
2
sin
2
(
x
2
)
x
2
d
x
I=\int_{-\infty}^{+\infty}\frac{e^{-x^{2}}\sin^{2}\left(x^{2}\right)}{x^{2}}\mathrm{d}x
I=∫−∞+∞x2e−x2sin2(x2)dx
由于是偶函数,所以:
I
=
2
∫
0
+
∞
e
−
x
2
sin
2
(
x
2
)
x
2
d
x
I=2\int_{0}^{+\infty}\frac{e^{-x^{2}}\sin^{2}\left(x^{2}\right)}{x^{2}}\mathrm{d}x
I=2∫0+∞x2e−x2sin2(x2)dx
下面使用一个小技巧,即通过添加参数
t
t
t 拓展上述积分:
I
(
t
)
=
2
∫
0
+
∞
e
−
x
2
sin
2
(
t
x
2
)
x
2
d
x
I(t)=2\int_{0}^{+\infty}\frac{e^{-x^{2}}\sin^{2}\left(tx^{2}\right)}{x^{2}}\mathrm{d}x
I(t)=2∫0+∞x2e−x2sin2(tx2)dx
然后对上式等号左右进行微分:
d
d
t
I
(
t
)
=
d
d
t
2
∫
0
+
∞
e
−
x
2
sin
2
(
t
x
2
)
x
2
d
x
=
2
∫
0
+
∞
∂
∂
t
e
−
x
2
sin
2
(
t
x
2
)
x
2
d
x
=
2
∫
0
+
∞
e
−
x
2
x
2
2
sin
(
t
x
2
)
cos
(
t
x
2
)
x
2
d
x
=
4
∫
0
+
∞
e
−
x
2
sin
(
t
x
2
)
cos
(
t
x
2
)
d
x
=
4
∫
0
+
∞
e
−
x
2
sin
(
2
t
x
2
)
d
x
\begin{aligned} \frac{\mathrm{d}}{\mathrm{d}t}I(t)&=\frac{\mathrm{d}}{\mathrm{d}t}2\int_{0}^{+\infty}\frac{e^{-x^{2}}\sin^{2}\left(tx^{2}\right)}{x^{2}}\mathrm{d}x\\ &=2\int_{0}^{+\infty}\frac{\partial}{\partial t}\frac{e^{-x^{2}}\sin^{2}\left(tx^{2}\right)}{x^{2}}\mathrm{d}x\\ &=2\int_{0}^{+\infty}\frac{e^{-x^{2}}}{x^{2}}2\sin\left(tx^{2}\right)\cos\left(tx^{2}\right)x^{2}\mathrm{d}x\\ &=4\int_{0}^{+\infty}e^{-x^{2}}\sin\left(tx^{2}\right)\cos\left(tx^{2}\right)\mathrm{d}x\\ &=4\int_{0}^{+\infty}e^{-x^{2}}\sin\left(2tx^{2}\right)\mathrm{d}x\\ \end{aligned}
dtdI(t)=dtd2∫0+∞x2e−x2sin2(tx2)dx=2∫0+∞∂t∂x2e−x2sin2(tx2)dx=2∫0+∞x2e−x22sin(tx2)cos(tx2)x2dx=4∫0+∞e−x2sin(tx2)cos(tx2)dx=4∫0+∞e−x2sin(2tx2)dx
由于:
e
i
x
=
cos
x
+
i
sin
x
e^{\mathrm{i}x}=\cos x+\mathrm{i}\sin x
eix=cosx+isinx
所以:
I
m
[
e
2
i
t
x
2
]
=
sin
(
2
t
x
2
)
\mathrm{Im}\left[e^{2\mathrm{i}tx^{2}}\right]=\sin\left(2tx^{2}\right)
Im[e2itx2]=sin(2tx2)
代入积分方程中:
I
′
(
t
)
=
I
m
[
4
∫
0
+
∞
e
−
x
2
e
2
i
t
x
2
d
x
]
=
I
m
[
4
∫
0
+
∞
e
−
x
2
(
1
−
2
i
t
)
d
x
]
\begin{aligned} I'(t) &=\mathrm{Im}\left[4\int_{0}^{+\infty}e^{-x^{2}}e^{2\mathrm{i}tx^{2}}\mathrm{d}x\right]\\ &=\mathrm{Im}\left[4\int_{0}^{+\infty}e^{-x^{2}(1-2\mathrm{i}t)}\mathrm{d}x\right]\\ \end{aligned}
I′(t)=Im[4∫0+∞e−x2e2itx2dx]=Im[4∫0+∞e−x2(1−2it)dx]
考虑到:
∫
0
+
∞
e
−
α
x
2
d
x
=
1
2
π
α
\int_{0}^{+\infty}e^{-\alpha x^{2}}\mathrm{d}x=\frac{1}{2}\sqrt{\frac{\pi}{\alpha}}
∫0+∞e−αx2dx=21απ
代入到前式中:
I
′
(
t
)
=
2
π
I
m
[
1
1
−
2
i
t
]
\begin{aligned} I'(t) &=2\sqrt{\pi}\mathrm{Im}\left[\frac{1}{\sqrt{1-2\mathrm{i}t}}\right]\\ \end{aligned}
I′(t)=2πIm[1−2it1]
将上式等号两端积分:
∫
0
+
∞
d
d
t
I
(
t
)
d
t
=
I
(
t
)
=
2
π
I
m
[
∫
0
+
∞
(
1
−
2
i
t
)
−
1
/
2
d
t
]
=
2
π
I
m
[
(
1
−
2
i
t
)
1
/
2
1
2
(
−
2
i
)
+
C
]
=
2
π
I
m
[
i
(
1
−
2
i
t
)
1
/
2
]
+
C
\begin{aligned} \int_{0}^{+\infty}\frac{\mathrm{d}}{\mathrm{d}t}I(t)\mathrm{d}t &=I(t)\\ &=2\sqrt{\pi}\mathrm{Im}\left[\int_{0}^{+\infty}\left(1-2\mathrm{i}t\right)^{-1/2}\mathrm{d}t\right]\\ &=2\sqrt{\pi}\mathrm{Im}\left[\frac{(1-2\mathrm{i}t)^{1/2}}{\frac{1}{2}(-2\mathrm{i})}+C\right]\\ &=2\sqrt{\pi}\mathrm{Im}\left[\mathrm{i}(1-2\mathrm{i}t)^{1/2}\right]+C\\ \end{aligned}
∫0+∞dtdI(t)dt=I(t)=2πIm[∫0+∞(1−2it)−1/2dt]=2πIm[21(−2i)(1−2it)1/2+C]=2πIm[i(1−2it)1/2]+C
考虑到:
I
(
t
)
=
2
∫
0
+
∞
e
−
x
2
sin
2
(
t
x
2
)
x
2
d
x
I(t)=2\int_{0}^{+\infty}\frac{e^{-x^{2}}\sin^{2}\left(tx^{2}\right)}{x^{2}}\mathrm{d}x
I(t)=2∫0+∞x2e−x2sin2(tx2)dx
此时,令
t
=
0
t=0
t=0,
I
(
t
=
0
)
=
0
I(t=0)=0
I(t=0)=0
则前式的结果:
I
(
t
=
0
)
=
0
=
2
π
I
m
[
i
(
1
−
2
i
t
)
1
/
2
]
+
C
=
2
π
I
m
[
i
]
+
C
=
2
π
+
C
\begin{aligned} I(t=0) &=0\\ &=2\sqrt{\pi}\mathrm{Im}\left[\mathrm{i}(1-2\mathrm{i}t)^{1/2}\right]+C\\ &=2\sqrt{\pi}\mathrm{Im}\left[\mathrm{i}\right]+C\\ &=2\sqrt{\pi}+C\\ \end{aligned}
I(t=0)=0=2πIm[i(1−2it)1/2]+C=2πIm[i]+C=2π+C
由上得出:
C
=
−
2
π
C=-2\sqrt{\pi}
C=−2π
则最初需要解决的积分:
I
(
t
=
1
)
=
∫
−
∞
+
∞
e
−
x
2
sin
2
(
x
2
)
x
2
d
x
=
2
π
I
m
[
i
(
1
−
2
i
)
1
/
2
]
−
2
π
\begin{aligned} I(t=1) &=\int_{-\infty}^{+\infty}\frac{e^{-x^{2}}\sin^{2}\left(x^{2}\right)}{x^{2}}\mathrm{d}x\\ &=2\sqrt{\pi}\mathrm{Im}\left[\mathrm{i}(1-2\mathrm{i})^{1/2}\right]-2\sqrt{\pi}\\ \end{aligned}
I(t=1)=∫−∞+∞x2e−x2sin2(x2)dx=2πIm[i(1−2i)1/2]−2π
设:
z
=
1
−
2
i
z=1-2\mathrm{i}
z=1−2i
则:
∣
z
∣
=
1
2
+
2
2
=
5
A
r
g
z
=
tan
−
1
(
−
2
1
)
=
−
tan
−
1
(
2
)
\left|z\right|=\sqrt{1^{2}+2^{2}}=\sqrt{5}\\ \mathrm{Arg}\ z=\tan^{-1}\left(\frac{-2}{1}\right)=-\tan^{-1}(2)
∣z∣=12+22=5Argz=tan−1(1−2)=−tan−1(2)
所以:
z
=
1
−
2
i
=
5
e
−
i
tan
−
1
(
2
)
z=1-2\mathrm{i}=\sqrt{5}e^{-\mathrm{i}\tan^{-1}(2)}
z=1−2i=5e−itan−1(2)
所以:
1
−
2
i
=
5
e
−
i
tan
−
1
(
2
)
2
\sqrt{1-2\mathrm{i}}=\sqrt{\sqrt{5}}e^{-\mathrm{i}\frac{\tan^{-1}(2)}{2}}
1−2i=5e−i2tan−1(2)
所以:
I
m
[
i
1
−
2
i
]
=
I
m
[
i
5
e
−
i
tan
−
1
(
2
)
2
]
=
5
cos
(
tan
−
1
(
2
)
2
)
\begin{aligned} \mathrm{Im}\left[ \mathrm{i}\sqrt{1-2\mathrm{i}}\right] &=\mathrm{Im}\left[ \mathrm{i}\sqrt{\sqrt{5}}e^{-\mathrm{i}\frac{\tan^{-1}(2)}{2}}\right]\\ &=\sqrt{\sqrt{5}}\cos\left(\frac{\tan^{-1}(2)}{2}\right) \end{aligned}
Im[i1−2i]=Im[i5e−i2tan−1(2)]=5cos(2tan−1(2))