数学代写 | MATH0065 Application of Complex

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=reiθ =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(iθ) 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 = 1−i, z = −1−i, 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 ∆z→0

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 ∆z→0

( z 0 + ∆ z ) 2 − z 02 ∆z

∆z→0

∆z

= lim(2z0+∆z)=2z0, ∆z→0