日: 2013年12月26日

Complex numbers

Complex numbers arose in order to solve polynomial equations such as x^2 + 1 = 0 or x^2 + x + 1 = 0 which are not satisfied by real numbers. It’s assumed that a complex number has the form a + bi where a, b are real numbers and i, called imaginary unit, has the property that i2 = -1. Complex numbers are defined as follows.

  1. Addition.
    \displaystyle (a + bi) + (c + di) = (a + c) + (b + d)i
  2. Subtraction.
    \displaystyle (a + bi) - (c + di) = (a - c) + (b - d)i
  3. Multiplication.
    \displaystyle (a + bi)\times(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i
  4. Division.
    \displaystyle \frac{a + bi}{c + di} = \frac{a + bi}{c + di}\times\frac{c - di}{c - di} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2}i

The ordinary rules of algebra has been used except that replaces i2 by -1 wherever it occurs. The commutative, associative and distributive laws also apply to complex numbers. It’s called a and b of a + bi the real and imaginary parts, respectively. Two complex numbers are equal if and only if their real and imaginary parts are respectively equal.

A complex number z = x + iy can be considered as a point P with coordinates (x, y) on a rectangular xy plane called in this case the complex plane or Argand diagram. If the line would be constructed from origin O to P and let ρ be the distance OP and φ the angle made by OP with the positive x axis, you could have from Figure

Graph

\displaystyle x = \rho \cos\phi,\ y = \rho\sin\phi,\ \rho = \sqrt{x^2 + y^2}

and could write the complex number in so-called polar form as

\displaystyle z = x + iy = \rho(\cos\phi + i\sin\phi) = \rho cis\phi

It’s often called that ρ the modulus or absolute value of of z and denote it by |z|. The angle φ is called the amplitude or argument of z abbreviated arg z. It could be also written \rho = \sqrt{z\bar{z}} where \bar{z} = x - iy is called the conjugate of z = x + iy.

If you write two complex numbers in polar form as

\displaystyle z_1 = \rho_1(\cos\phi_1 + i\sin\phi_1),\ z_2 = \rho_2(\cos\phi_2 + i\sin\phi_2)

then

\displaystyle z_1z_2 = \rho_1\rho_2[\cos(\phi_1 + \phi_2) + i\sin(\phi_1 + \phi_2)]\\\vspace{0.2 in} \frac{z_1}{z_2} = \frac{\rho_1}{\rho_2}[\cos(\phi_1 - \phi_2) + i\sin(\phi_1 - \phi_2)]

Also if n is any real number, you have

\displaystyle z^n = [\rho(\cos\phi + i\sin\phi)]^n = \rho^n(\cos n\phi + i\sin n\phi)

which is often called De Moivre’s theorem. You can use this to determine roots of complex numbers. For example if n is a positive integer,

\displaystyle z^{\frac{1}{n}} = [\rho(\cos\phi + i\sin\phi)]^\frac{1}{n} = \rho^{\frac{1}{n}}\left\{\cos\left(\frac{\phi + 2k\pi}{n}\right) + i\sin\left(\frac{\phi + 2k\pi}{n}\right)\right\}\\\vspace{0.2 in} \ \ k = 0,\ 1,\ 2,\ \cdots,\ n-1

Using the series for ex, sin x, cos x, you are led to define

\displaystyle e^{i\phi} = \cos\phi + i\sin\phi,\ e^{-i\phi} = \cos\phi - i\sin\phi

which are called Euler’s formulas and which enable you to rewrite equations in terms of exponentials.

複素数

　複素数は x^2 + 1 = 0 や x^2 + x + 1 = 0 といった実数解を持たない整方程式を解く際に出現します．複素数は a + bi という形をしており，a, b は実数部であり，i は虚数部と呼ばれ，i2 = -1 という性質を持ちます．複素数は以下のように定義されます．

  1. 加算
    \displaystyle (a + bi) + (c + di) = (a + c) + (b + d)i
  2. 減算
    \displaystyle (a + bi) - (c + di) = (a - c) + (b - d)i
  3. 乗算
    \displaystyle (a + bi)\times(c + di) = ac + adi + bci + bdi^2 = (ac - bd) + (ad + bc)i
  4. 除算
    \displaystyle \frac{a + bi}{c + di} = \frac{a + bi}{c + di}\times\frac{c - di}{c - di} = \frac{ac + bd}{c^2 + d^2} + \frac{bc - ad}{c^2 + d^2}i

　いかなる状況でも i2 を -1 で置換することを除けば，代数の通常の定義が用いられます．交換法則，結合法則，分配法則もまた複素数に適用されます．a + bi の a および b はそれぞれ 実数部 および 虚数部 と呼ばれます．二つの複素数が等しいとは，実数部と虚数部とがそれぞれ等しい時に限られます．

　ある複素数 z = x + iy は xy 直交平面上の (x, y) 座標の点 P とみなすことができ，この場合この平面を 複素平面 または アルガン図 と呼びます．仮に原点 O から点 P への直線を引き，OP 間の距離を ρ とし，OP と x 軸のなす角度を φ とすると，次の図を得ます．

Graph

\displaystyle x = \rho \cos\phi,\ y = \rho\sin\phi,\ \rho = \sqrt{x^2 + y^2}

また複素数をいわゆる 極座標系式 として以下のように記述できます．

\displaystyle z = x + iy = \rho(\cos\phi + i\sin\phi) = \rho cis\phi

　しばしば ρ は z の モジュラス または 絶対値 と呼ばれ，|z| と記述します．角 φ は z の amplitude または argument と呼ばれ，arg z と略記します．\rho = \sqrt{z\bar{z}} とも記述できます．ただし \bar{z} = x - iy は z = x + iy の 共役複素数 と呼ばれます．

　極座標で複素数を記述すると以下のようになります．

\displaystyle z_1 = \rho_1(\cos\phi_1 + i\sin\phi_1),\ z_2 = \rho_2(\cos\phi_2 + i\sin\phi_2)

すると

\displaystyle z_1z_2 = \rho_1\rho_2[\cos(\phi_1 + \phi_2) + i\sin(\phi_1 + \phi_2)]\\\vspace{0.2 in} \frac{z_1}{z_2} = \frac{\rho_1}{\rho_2}[\cos(\phi_1 - \phi_2) + i\sin(\phi_1 - \phi_2)]

　仮に n を任意の実数とすると以下を得ます．

\displaystyle z^n = [\rho(\cos\phi + i\sin\phi)]^n = \rho^n(\cos n\phi + i\sin n\phi)

これは ド・モアブルの定理 として知られています．これを用いて複素数の根を定義できます．例えば仮に n を正の整数と仮定すると，

\displaystyle z^{\frac{1}{n}} = [\rho(\cos\phi + i\sin\phi)]^\frac{1}{n} = \rho^{\frac{1}{n}}\left\{\cos\left(\frac{\phi + 2k\pi}{n}\right) + i\sin\left(\frac{\phi + 2k\pi}{n}\right)\right\}\\\vspace{0.2 in} \ \ k = 0,\ 1,\ 2,\ \cdots,\ n-1

ex, sin x, cos x のための級数を用いると以下の定義に導かれます．

\displaystyle e^{i\phi} = \cos\phi + i\sin\phi,\ e^{-i\phi} = \cos\phi - i\sin\phi

これらは オイラーの公式 と呼ばれ，方程式を指数関数の観点から書き直すことができます．