附录5-6_复数与积分表

附录5 复数及其运算

一、复数的概念

我们知道在实数范围内,方程

x ^ {2} = - 1

是无解的,因为没有一个实数的平方等于-1. 由于解方程的需要,人们引进一个新数 $i$ ,称为虚数单位,并规定

\mathrm{i} ^ {2} = - 1,

从而 $i$ 是方程 $x^{2}=-1$ 的一个根.

对于任意两个实数 $x, y$ ，我们称 $z = x + yi$ 为复数，其中 $x, y$ 分别称为 $z$ 的实部和虚部，记作

x = \operatorname{Re} (z), \quad y = \operatorname{Im} (z).

当 $x = 0, y \neq 0$ 时， $z = \mathrm{i}y$ 称为纯虚数；当 $y = 0$ 时， $z = x + 0\mathrm{i}$ ，我们把它看作是实数 $x$ .

两个复数相等,必须且只需它们的实部和虚部分别相等.一个复数 $z$ 等于0,必须且只需它的实部和虚部同时等于0.

二、复数的代数运算

两个复数 $z_{1} = x_{1} + \mathrm{i}y_{1}, z_{2} = x_{2} + \mathrm{i}y_{2}$ 的加法、减法及乘法定义如下：

\left(x _ {1} + \mathrm{i} y _ {1}\right) \pm \left(x _ {2} + \mathrm{i} y _ {2}\right) = \left(x _ {1} \pm x _ {2}\right) + \mathrm{i} \left(y _ {1} \pm y _ {2}\right),

\left(x _ {1} + \mathrm{i} y _ {1}\right) \left(x _ {2} + \mathrm{i} y _ {2}\right) = \left(x _ {1} x _ {2} - y _ {1} y _ {2}\right) + \mathrm{i} \left(x _ {2} y _ {1} + x _ {1} y _ {2}\right),

并分别称以上两式右端的复数为 $z_{1}$ 与 $z_{2}$ 的和、差与积.

我们又称满足

z _ {2} z = z _ {1} \quad (z _ {2} \neq 0)

的复数 $z = x + \mathrm{i}y$ 为 $z_{1}$ 除以 $z_{2}$ 的商，记作 $z = \frac{z_1}{z_2}$ ，从这个定义立即可推得

z = \frac {z _ {1}}{z _ {2}} = \frac {x _ {1} x _ {2} + y _ {1} y _ {2}}{x _ {2} ^ {2} + y _ {2} ^ {2}} + \mathrm{i} \frac {x _ {2} y _ {1} - x _ {1} y _ {2}}{x _ {2} ^ {2} + y _ {2} ^ {2}}.

不难证明,与实数的情形一样,复数的运算也满足交换律、结合律和分配律

z _ {1} + z _ {2} = z _ {2} + z _ {1}, \quad z _ {1} z _ {2} = z _ {2} z _ {1},

\begin{array}{r l} z _ {1} + (z _ {2} + z _ {3}) & = (z _ {1} + z _ {2}) + z _ {3}, \quad z _ {1} (z _ {2} z _ {3}) = (z _ {1} z _ {2}) z _ {3}, \\ & \quad - z _ {1} (z _ {2} + z _ {3}) = z _ {1} z _ {2} + z _ {1} z _ {3}. \end{array}

我们把实部相同而虚部绝对值相等符号相反的两个复数称为共轭复数，与 $z$ 共轭的复数记为 $\bar{z}$ 。如果 $z = x + \mathrm{i}y$ ，那么 $\bar{z} = x - \mathrm{i}y$ 。共轭复数有如下性质：

$\overline{z_1 \pm z_2} = \overline{z_1} \pm \overline{z_2}$ , $\overline{z_1 z_2} = \overline{z_1} \overline{z_2}$ , $\left(\frac{z_1}{z_2}\right) = \frac{\overline{z_1}}{\overline{z_2}}$ .
$z \bar{z} = [\operatorname{Re}(z)]^{2} + [\operatorname{Im}(z)]^{2}$ .
$z+\bar{z}=2\mathrm{Re}(z)$ , $z-\bar{z}=2\mathrm{i}\mathrm{Im}(z)$ .

三、复数的三种表示式及 Euler 公式

代数表示式

z = x + \mathrm{i} y.

三角表示式

z = r (\cos \theta + \mathrm{i} \sin \theta).

指数表示式

z = r \mathrm{e} ^ {\mathrm{i} \theta}.

Euler 公式

\mathrm{e} ^ {\mathrm{i} \theta} = \cos \theta + \mathrm{i} \sin \theta ,

由此可得

\cos \theta = \frac {1}{2} (\mathrm{e} ^ {\mathrm{i} \theta} + \mathrm{e} ^ {- \mathrm{i} \theta}), \quad \sin \theta = \frac {1}{2 \mathrm{i}} (\mathrm{e} ^ {\mathrm{i} \theta} - \mathrm{e} ^ {- \mathrm{i} \theta}).

附录6 简明积分表

一、含有 $a + bx$ 的积分

1. \int (a + b x) ^ {n} \mathrm{d} x = \left\{ \begin{array}{l l} \frac {(a + b x) ^ {n + 1}}{b (n + 1)} + C, & \text {当} n \neq - 1, \\ \frac {1}{b} \ln | a + b x | + C, & \text {当} n = - 1. \end{array} \right.

$\int \frac{x\mathrm{d}x}{a + bx} = \frac{x}{b} -\frac{a}{b^2}\ln |a + bx| + C.$
$\int \frac{x^2\mathrm{d}x}{a + bx} = \frac{1}{b^3}\left[\frac{1}{2} (a + bx)^2 - 2a(a + bx) + a^2\ln |a + bx|\right] + C.$

\int \frac {x \mathrm{d} x}{(a + b x) ^ {2}} = \frac {1}{b ^ {2}} \left(\frac {a}{a + b x} + \ln | a + b x |\right) + C.

\int \frac {x ^ {2} \mathrm{d} x}{(a + b x) ^ {2}} = \frac {x}{b ^ {2}} - \frac {a ^ {2}}{b ^ {3} (a + b x)} - \frac {2 a}{b ^ {3}} \ln | a + b x | + C.

$\int \frac{\mathrm{d}x}{x(a + bx)} = \frac{1}{a}\ln \left|\frac{x}{a + bx}\right| + C.$
$\int \frac{\mathrm{d}x}{x^2(a + bx)} = -\frac{1}{ax} +\frac{b}{a^2}\ln \left|\frac{a + bx}{x}\right| + C.$

\int \frac {\mathrm{d} x}{x (a + b x) ^ {2}} = \frac {1}{a (a + b x)} - \frac {1}{a ^ {2}} \ln \left| \frac {a + b x}{x} \right| + C.

二、含有 $\sqrt{a + bx}$ 的积分

$\int x\sqrt{a + bx}\mathrm{d}x = \frac{2(3bx - 2a)(a + bx)^{3 / 2}}{15b^2} +C.$
$\int x^{2}\sqrt{a + bx}\mathrm{d}x = \frac{2(15b^{2}x^{2} - 12abx + 8a^{2})(a + bx)^{3 / 2}}{105b^{3}} +C.$
$\int \frac{x\mathrm{d}x}{\sqrt{a + bx}} = \frac{2(bx - 2a)\sqrt{a + bx}}{3b^2} + C.$
$\int \frac{x^2\mathrm{d}x}{\sqrt{a + bx}} = \frac{2(3b^2x^2 - 4abx + 8a^2)\sqrt{a + bx}}{15b^3} + C.$
$\int \frac{\mathrm{d}x}{x\sqrt{a + bx}} = \left\{ \begin{array}{ll} \frac{1}{\sqrt{a}}\ln \frac{|\sqrt{a + bx} - \sqrt{a}|}{\sqrt{a + bx} + \sqrt{a}} + C, & \text{当} a > 0,\\ \frac{2}{\sqrt{-a}}\arctan \sqrt{\frac{a + bx}{-a}} +C, & \text{当} a < 0. \end{array} \right.$

\int \frac {\mathrm{d} x}{x ^ {2} \sqrt {a + b x}} = - \frac {\sqrt {a + b x}}{a x} - \frac {b}{2 a} \int \frac {\mathrm{d} x}{x \sqrt {a + b x}}.

\int \frac {\sqrt {a + b x}}{x} \mathrm{d} x = 2 \sqrt {a + b x} + a \int \frac {\mathrm{d} x}{x \sqrt {a + b x}}.

$\int \frac{\sqrt{a + bx}}{x^2}\mathrm{d}x = -\frac{\sqrt{a + bx}}{x} +\frac{b}{2}\int \frac{\mathrm{d}x}{x\sqrt{a + bx}}.$

三、含有 $a^2 \pm x^2$ 的积分

$\int \frac{\mathrm{d}x}{(a^2 + x^2)^n} = \begin{cases} \frac{1}{a}\arctan \frac{x}{a} + C, & \text{当 } n = 1, \\ \frac{x}{2(n - 1)a^2(a^2 + x^2)^{n - 1}} + \frac{2n - 3}{2(n - 1)a^2}\int \frac{\mathrm{d}x}{(a^2 + x^2)^{n - 1}}, & \text{当 } n > 1. \end{cases}$
$\int \frac{x\mathrm{d}x}{(a^2 + x^2)^n} = \begin{cases} \frac{1}{2}\ln (a^2 + x^2) + C, & \text{当 } n = 1, \\ -\frac{1}{2(n - 1)(a^2 + x^2)^{n - 1}} + C, & \text{当 } n > 1. \end{cases}$
$\int \frac{\mathrm{d}x}{a^2 - x^2} = \frac{1}{2a}\ln \left|\frac{a + x}{a - x}\right| + C.$

四、含有 $\sqrt{a^2 - x^2}$ 的积分

$\int \sqrt{a^2 - x^2} \, \mathrm{d}x = \frac{x}{2}\sqrt{a^2 - x^2} + \frac{a^2}{2}\arcsin \frac{x}{a} + C.$
$\int x\sqrt{a^2 - x^2}\mathrm{d}x = -\frac{1}{3} (a^2 -x^2)^{3 / 2} + C.$
$\int x^{2}\sqrt{a^{2} - x^{2}}\mathrm{d}x = \frac{x}{8} (2x^{2} - a^{2})\sqrt{a^{2} - x^{2}} +\frac{a^{4}}{8}\arcsin \frac{x}{a} +C.$
$\int \frac{\mathrm{d}x}{\sqrt{a^2 - x^2}} = \arcsin \frac{x}{a} + C.$
$\int \frac{x\mathrm{d}x}{\sqrt{a^2 - x^2}} = -\sqrt{a^2 - x^2} + C.$
$\int \frac{x^2\mathrm{d}x}{\sqrt{a^2 - x^2}} = -\frac{x}{2}\sqrt{a^2 - x^2} +\frac{a^2}{2}\arcsin \frac{x}{a} +C.$

\int \left(a ^ {2} - x ^ {2}\right) ^ {3 / 2} \mathrm{d} x = \frac {x}{8} \left(5 a ^ {2} - 2 x ^ {2}\right) \sqrt {a ^ {2} - x ^ {2}} + \frac {3 a ^ {4}}{8} \arcsin \frac {x}{a} + C.

$\int \frac{\mathrm{d}x}{(a^2 - x^2)^{3 / 2}} = \frac{x}{a^2\sqrt{a^2 - x^2}} + C.$
$\int \frac{x\mathrm{d}x}{(a^2 - x^2)^{3 / 2}} = \frac{1}{\sqrt{a^2 - x^2}} + C.$
$\int \frac{x^2\mathrm{d}x}{(a^2 - x^2)^{3 / 2}} = \frac{x}{\sqrt{a^2 - x^2}} -\arcsin \frac{x}{a} +C.$

\int \frac {\mathrm{d} x}{x \sqrt {a ^ {2} - x ^ {2}}} = \frac {1}{a} \ln \left| \frac {a - \sqrt {a ^ {2} - x ^ {2}}}{x} \right| + C.

$\int \frac{\mathrm{d}x}{x^2\sqrt{a^2 - x^2}} = -\frac{\sqrt{a^2 - x^2}}{a^2x} + C.$

\int \frac {\mathrm{d} x}{x ^ {3} \sqrt {a ^ {2} - x ^ {2}}} = - \frac {\sqrt {a ^ {2} - x ^ {2}}}{2 a ^ {2} x ^ {2}} - \frac {1}{2 a ^ {3}} \ln \left| \frac {a + \sqrt {a ^ {2} - x ^ {2}}}{x} \right| + C.

$\int \frac{\sqrt{a^2 - x^2}}{x} \mathrm{d}x = \sqrt{a^2 - x^2} - a\ln \left|\frac{a + \sqrt{a^2 - x^2}}{x}\right| + C.$
$\int \frac{\sqrt{a^2 - x^2}}{x^2}\mathrm{d}x = -\frac{\sqrt{a^2 - x^2}}{x} -\arcsin \frac{x}{a} +C.$

五、含有 $\sqrt{x^2\pm a^2}$ 的积分

$\int \sqrt{x^2\pm a^2}\mathrm{d}x = \frac{x}{2}\sqrt{x^2\pm a^2}\pm \frac{a^2}{2}\ln |x + \sqrt{x^2\pm a^2}| + C.$
$\int x\sqrt{x^2\pm a^2}\mathrm{d}x = \frac{1}{3} (x^2\pm a^2)^{3 / 2} + C.$
$\int x^{2}\sqrt{x^{2} \pm a^{2}}\mathrm{d}x = \frac{x}{8}(2x^{2} \pm a^{2})\sqrt{x^{2} \pm a^{2}} - \frac{a^{4}}{8}\ln |x + \sqrt{x^{2} \pm a^{2}}| + C.$
$\int \frac{\mathrm{d}x}{\sqrt{x^2\pm a^2}} = \ln |x + \sqrt{x^2\pm a^2}| + C.$
$\int \frac{x\mathrm{d}x}{\sqrt{x^2\pm a^2}} = \sqrt{x^2\pm a^2} +C.$
$\int \frac{x^2\mathrm{d}x}{\sqrt{x^2\pm a^2}} = \frac{x}{2}\sqrt{x^2\pm a^2}\mp \frac{a^2}{2}\ln |x + \sqrt{x^2\pm a^2}| + C.$
$\int (x^{2} \pm a^{2})^{3/2} \mathrm{d}x = \frac{x}{8}(2x^{2} \pm 5a^{2})\sqrt{x^{2} \pm a^{2}} + \frac{3a^{4}}{8}\ln |x + \sqrt{x^{2} \pm a^{2}}| + C.$
$\int \frac{\mathrm{d}x}{(x^2\pm a^2)^{3 / 2}} = \pm \frac{x}{a^2\sqrt{x^2\pm a^2}} +C.$
$\int \frac{x\mathrm{d}x}{(x^2\pm a^2)^{3 / 2}} = -\frac{1}{\sqrt{x^2\pm a^2}} +C.$
$\int \frac{x^2\mathrm{d}x}{(x^2\pm a^2)^{3 / 2}} = -\frac{x}{\sqrt{x^2\pm a^2}} +\ln |x + \sqrt{x^2\pm a^2}| + C.$
$\int \frac{\mathrm{d}x}{x^2\sqrt{x^2\pm a^2}} = \mp \frac{\sqrt{x^2\pm a^2}}{a^2x} +C.$
$\int \frac{\mathrm{d}x}{x^3\sqrt{x^2 + a^2}} = -\frac{\sqrt{x^2 + a^2}}{2a^2x^2} +\frac{1}{2a^3}\ln \frac{a + \sqrt{x^2 + a^2}}{|x|} +C.$
$\int \frac{\mathrm{d}x}{x^3\sqrt{x^2 - a^2}} = \frac{\sqrt{x^2 - a^2}}{2a^2x^2} +\frac{1}{2a^3}\arccos \frac{a}{x} +C.$
$\int \frac{\sqrt{x^2 + a^2}}{x}\mathrm{d}x = \sqrt{x^2 + a^2} -a\ln \frac{a + \sqrt{x^2 + a^2}}{|x|} +C.$
$\int \frac{\sqrt{x^2 - a^2}}{x} \, \mathrm{d}x = \sqrt{x^2 - a^2} - a\arccos \frac{a}{|x|} + C.$
$\int \frac{\sqrt{x^2\pm a^2}}{x^2}\mathrm{d}x = -\frac{\sqrt{x^2\pm a^2}}{x} +\ln |x + \sqrt{x^2\pm a^2}| + C.$

\int \frac {\mathrm{d} x}{x \sqrt {x ^ {2} + a ^ {2}}} = \frac {1}{a} \ln \frac {| x |}{a + \sqrt {x ^ {2} + a ^ {2}}} + C.

\int \frac {\mathrm{d} x}{x \sqrt {x ^ {2} - a ^ {2}}} = \frac {1}{a} \arccos \frac {a}{| x |} + C.

六、含有 $a + bx + cx^2$ 的积分

5 3. \int \frac {\mathrm{d} x}{a + b x + c x ^ {2}} = \left\{ \begin{array}{l l} \frac {2}{\sqrt {4 a c - b ^ {2}}} \arctan \frac {2 c x + b}{\sqrt {4 a c - b ^ {2}}} + C, & \text {当} b ^ {2} < 4 a c, \\ \frac {1}{\sqrt {b ^ {2} - 4 a c}} \ln \left| \frac {\sqrt {b ^ {2} - 4 a c} - b - 2 c x}{\sqrt {b ^ {2} - 4 a c} + b + 2 c x} \right| + C, & \text {当} b ^ {2} > 4 a c. \end{array} \right.

七、含有 $\sqrt{a + bx + cx^2}$ 的积分

5 4. \int \frac {\mathrm{d} x}{\sqrt {a + b x + c x ^ {2}}} = \left\{ \begin{array}{l l} \frac {1}{\sqrt {c}} \ln | 2 c x + b + 2 \sqrt {c (a + b x + c x ^ {2})} | + C, & \text {当} c > 0, \\ - \frac {1}{\sqrt {- c}} \arcsin \frac {2 c x + b}{\sqrt {b ^ {2} - 4 a c}} + C, & \text {当} c < 0, b ^ {2} > 4 a c. \end{array} \right.

\int \sqrt {a + b x + c x ^ {2}} \mathrm{d} x = \frac {2 c x + b}{4 c} \sqrt {a + b x + c x ^ {2}} + \frac {4 a c - b ^ {2}}{8 a} \int \frac {\mathrm{d} x}{\sqrt {a + b x + c x ^ {2}}}.

\int \frac {x \mathrm{d} x}{\sqrt {a + b x + c x ^ {2}}} = \frac {1}{c} \sqrt {a + b x + c x ^ {2}} - \frac {b}{2 c} \int \frac {\mathrm{d} x}{\sqrt {a + b x + c x ^ {2}}}.

八、含有三角函数的积分

\int \sin^ {2} a x \mathrm{d} x = \frac {1}{2 a} (a x - \sin a x \cos a x) + C.

\int \cos^ {2} a x \mathrm{d} x = \frac {1}{2 a} (a x + \sin a x \cos a x) + C.

\int \sin^ {n} x \mathrm{d} x = - \frac {1}{n} \sin^ {n - 1} x \cos x + \frac {n - 1}{n} \int \sin^ {n - 2} x \mathrm{d} x.

\int \cos^ {n} x \mathrm{d} x = \frac {1}{n} \cos^ {n - 1} x \sin x + \frac {n - 1}{n} \int \cos^ {n - 2} x \mathrm{d} x.

$\int \tan x\mathrm{d}x = -\ln |\cos x| + C.$
$\int \cot x\mathrm{d}x = \ln |\sin x| + C.$
$\int \tan^n x\mathrm{d}x = \frac{\tan^{n - 1}x}{n - 1} -\int \tan^{n - 2}x\mathrm{d}x.$
$\int \cot^n x\mathrm{d}x = -\frac{\cot^{n - 1}x}{n - 1} -\int \cot^{n - 2}x\mathrm{d}x.$
$\int \sec x\mathrm{d}x = \ln |\sec x + \tan x| + C.$
$\int \csc x\mathrm{d}x = -\ln |\csc x + \cot x| + C.$
$\int \sec^n x\mathrm{d}x = \frac{\tan x\sec^{n - 2}x}{n - 1} +\frac{n - 2}{n - 1}\int \sec^{n - 2}x\mathrm{d}x.$
$\int \csc^n x\mathrm{d}x = -\frac{\cot x\csc^{n - 2}x}{n - 1} +\frac{n - 2}{n - 1}\int \csc^{n - 2}x\mathrm{d}x.$
$\int \sec x\tan x\mathrm{d}x = \sec x + C.$
$\int \csc x\cot x\mathrm{d}x = -\csc x + C.$
$\int \sin ax\sin bx\mathrm{d}x = -\frac{\sin(a + b)x}{2(a + b)} +\frac{\sin(a - b)x}{2(a - b)} +C.$
$\int \sin ax\cos bx\mathrm{d}x = -\frac{\cos(a + b)x}{2(a + b)} -\frac{\cos(a - b)x}{2(a - b)} +C.$
$\int \cos ax\cos bx\mathrm{d}x = \frac{\sin(a + b)x}{2(a + b)} +\frac{\sin(a - b)x}{2(a - b)} +C.$

\begin{array}{r l} \int \sin^ {m} x \cos^ {n} x \mathrm{d} x & = \frac {\sin^ {m + 1} x \cos^ {n - 1} x}{m + n} + \frac {n - 1}{m + n} \int \sin^ {m} x \cos^ {n - 2} x \mathrm{d} x \\ & = - \frac {\sin^ {m - 1} x \cos^ {n + 1} x}{m + n} + \frac {m - 1}{m + n} \int \sin^ {m - 2} x \cos^ {n} x \mathrm{d} x. \end{array}

$\int \frac{\mathrm{d}x}{a + b\cos x} = \left\{ \begin{array}{ll}\frac{2}{\sqrt{a^2 - b^2}}\arctan \left(\sqrt{\frac{a - b}{a + b}}\tan \frac{x}{2}\right) + C, & \text{当} a^2 >b^2,\\ \frac{1}{\sqrt{b^2 - a^2}}\ln \left|\frac{b + a\cos x + \sqrt{b^2 - a^2}\sin x}{a + b\cos x}\right| + C, & \text{当} a^2 < b^2. \end{array} \right.$
$\int x\mathrm{e}^{x}\mathrm{d}x = \mathrm{e}^{x}(x - 1) + C.$
$\int x^{n}\mathrm{e}^{x}\mathrm{d}x = x^{n}\mathrm{e}^{x} - n\int x^{n - 1}\mathrm{e}^{x}\mathrm{d}x.$
$\int \ln x\mathrm{d}x = x(\ln x - 1) + C.$
$\int x^{n}\ln x\mathrm{d}x = \frac{x^{n + 1}}{(n + 1)^{2}} [(n + 1)\ln x - 1] + C.$

\int x \sin x \mathrm{d} x = \sin x - x \cos x + C.

\int x \cos x \mathrm{d} x = \cos x + x \sin x + C.

\int x ^ {n} \sin x \mathrm{d} x = - x ^ {n} \cos x + n \int x ^ {n - 1} \cos x \mathrm{d} x.

\int x ^ {n} \cos x \mathrm{d} x = x ^ {n} \sin x - n \int x ^ {n - 1} \sin x \mathrm{d} x.

8 4. \int \mathrm{e} ^ {a x} \sin b x \mathrm{d} x = \frac {\mathrm{e} ^ {a x} (a \sin b x - b \cos b x)}{a ^ {2} + b ^ {2}} + C.

8 5. \int \mathrm{e} ^ {a x} \cos b x \mathrm{d} x = \frac {\mathrm{e} ^ {a x} (a \cos b x + b \sin b x)}{a ^ {2} + b ^ {2}} + C.

\int \arcsin x \mathrm{d} x = x \arcsin x + \sqrt {1 - x ^ {2}} + C.

\int \arctan x \mathrm{d} x = x \arctan x - \frac {1}{2} \ln (1 + x ^ {2}) + C.

\int x ^ {n} \arcsin x \mathrm{d} x = \frac {1}{n + 1} \left(x ^ {n + 1} \arcsin x - \int \frac {x ^ {n + 1}}{\sqrt {1 - x ^ {2}}} \mathrm{d} x\right).

$\int x^{n}\arctan x\mathrm{d}x = \frac{1}{n + 1}\left(x^{n + 1}\arctan x - \int \frac{x^{n + 1}}{1 + x^{2}}\mathrm{d}x\right).$

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附录5-6_复数与积分表

附录5 复数及其运算

一、复数的概念

二、复数的代数运算

三、复数的三种表示式及 Euler 公式

附录6 简明积分表

一、含有 a+bx 的积分

二、含有 a+bx​ 的积分

三、含有 a2±x2 的积分

四、含有 a2−x2​ 的积分

五、含有 x2±a2​ 的积分

六、含有 a+bx+cx2 的积分

七、含有 a+bx+cx2​ 的积分