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z= x-iy。锻炼。验证| z | 2 = zz。 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（z） 然而 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） 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）= zz并观察到 lim（z0 + z）（z0+ z） z0z0 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 eect, 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 coecients. Dierentiation. The function f(z) is dierentiable 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