1复杂变量-简介/修订。
复数。我们写z = x + iy,其中x = Re(z),y = Im(z)是z的实部和虚部。

或者,使用复平面z中的极坐标,
z =reiθ= r(cosθ+isinθ),(1.1)

其中r = | z | =􏰅x2+ y2是z的绝对值(幅度,模数),而θ= arg(z)是z的自变量。该参数是一个多值函数的示例,因为对于任何整数n,exp [i(θ+2πn)] = exp(iθ)。当需要单值数量时,我们可以选择参数Arg(z)的主值,例如Arg(z)∈(-π,π]。沿切面的上侧包含负实轴且点z = x <0的复平面z。

锻炼。查找| z |和Arg(z)用于以下复数:z = 1 + i,z = 1-i,z = -1i,z = -1 + i。请注意,极角θ= arctan(y / x)的“标准”公式在这些计算中并不总是有效。

复数z = x + iy的复共轭为z̄= x-iy。锻炼。验证| z | 2 = zz̄。

功能。例如,我们写出w = f(z)= u(x,y)+ iv(x,y)。例如,对于许多标准函数,我们有f(z)= f(z̄)

然而

ez = ez̄,sinz = sinz,z3 = z3,(1 + i)z =(1 − i)z̄=(1 + i)z̄。

(1.2)

(1.3)

(1.4)

(1.5)(1.6)

(1.7)

显然,如果函数可以写成具有实系数的泰勒级数,则规则f(z)= f(z)适用。差异化。如果极限,函数f(z)在点z0上是可微的,

f′(z0)= limΔz→0

是否存在,并不取决于Δz趋于零的趋势。例子。令f(z)= z2。我们有

这是z2的一个熟悉的派生
例子。令f(z)= | z | 2。写下f(z)= zz̄并观察到

lim(z0 + ∆z)(z̄0+ ∆z)− z0z̄0

f(z0 +Δz)− f(z0),Δz

f(z0)= lim ∆z→0

(z 0 + ∆ z)2 − z 02 ∆z

Δz→0

Δz

Variables

1 Complex variables – introduction/revision.
Complex numbers. We write z = x+iy, where x = Re(z),y = Im(z) are the real and imaginary parts of z.

Alternatively, using polar coordinates in the complex plane z,
z
=re=r(cosθ+isinθ), (1.1)

where r = |z| = 􏰅x2 + y2 is the absolute value (magnitude, modulus) of z and θ = arg(z) is the argument of z. The argument is an example of a multi-valued function since exp[i(θ + 2πn)] = exp() for any integer n. When a single- valued quantity is required we can choose the principal value of the argument, Arg(z) such that Arg(z) (π, π], for example. In e􏰀ect, this creates a branch cut in the complex plane z along the negative real axis with the points z = x < 0 included on the upper side of the cut.

Exercise. Find |z| and Arg(z) for the following complex numbers: z = 1+i, z = 1i, z = 1i, z = 1+i. Note that the ‘standard’ formula for the polar angle θ = arctan(y/x) does not always work in these calculations.

The complex conjugate of a complex number z = x + iy is z ̄ = x iy. Exercise. Verify that |z|2 = zz ̄.

Functions. We write, for example, w = f(z) = u(x,y)+iv(x,y). For many standard functions we have f(z) = f(z ̄), for example

however

ez = ez ̄,sinz = sinz,z3 = z3, (1 + i)z = (1 i)z ̄ ̸= (1 + i)z ̄.

(1.2)

(1.3)

(1.4)

(1.5) (1.6)

(1.7)

Clearly the rule f(z) = f(z) applies if the function can be written as a Taylor series with real coe􏰂cients. Di􏰀erentiation. The function f(z) is di􏰀erentiable at some point z0 if the limit,

f(z0) = lim z0

exists and does not depend on how z tends to zero. Example. Let f(z) = z2. We have

which is a familiar derivative of z2.
Example. Let f(z) = |z|2. Write f(z) = zz ̄ and observe that

lim (z0 + ∆z)(z ̄0 + ∆z) z0z ̄0

f(z0 + ∆z) f(z0), z

f(z0)= lim z0

( z 0 + ∆ z ) 2 z 02 z

z0

z

= lim(2z0+∆z)=2z0, z0


EasyDue™ 支持PayPal, AliPay, WechatPay, Taobao等各种付款方式!

E-mail: easydue@outlook.com  微信:easydue


EasyDue™是一个服务全球中国留学生的专业代写公司
专注提供稳定可靠的北美、澳洲、英国代写服务
专注提供CS、统计、金融、经济、数学等覆盖100+专业的作业代写服务