Complex Variables

The imaginary unit in complex numbers is denoted by 'i' and is defined as the square root of -1, i.e., (i = \sqrt{-1}).

The real part of a complex number is the part without the imaginary unit 'i'. Therefore, the real part of (z = 3 + 4i) is 3.

The imaginary part of a complex number is the part that contains the imaginary unit 'i'. Therefore, the imaginary part of (z = 3 + 4i) is 4.

The complex conjugate of a complex number (z = a + bi) is (\bar{z} = a - bi). Therefore, the complex conjugate of (z = 3 + 4i) is (3 - 4i).

The modulus (absolute value) of a complex number (z = a + bi) is (|z| = \sqrt{a^2 + b^2}). Therefore, the modulus of (z = 3 + 4i) is (|z| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5).

The argument (phase) of a complex number (z = a + bi) is (\theta = \arctan(b/a)). Therefore, the argument of (z = 3 + 4i) is (\theta = \arctan(4/3)).

The polar form of a complex number (z = a + bi) is (z = r(\cos(\theta) + i\sin(\theta))), where (r = |z|) and (\theta = \arg(z)). Therefore, the polar form of (z = 3 + 4i) is (5(\cos(\arctan(4/3)) + i\sin(\arctan(4/3)))).

The exponential form of a complex number (z = a + bi) is (z = re^{i\theta}), where (r = |z|) and (\theta = \arg(z)). Therefore, the exponential form of (z = 3 + 4i) is (5e^{i\arctan(4/3)}).

The derivative of a complex function (f(z)) is defined as (f'(z) = \lim_{h\to 0} \frac{f(z+h) - f(z)}{h}). Using this definition, we can find that the derivative of (f(z) = z^2) is (f'(z) = 2z). Therefore, the derivative of (f(z) = z^2) at (z = 2) is (f'(2) = 2(2) = 4).

The Cauchy-Riemann equations are a system of two partial differential equations that a complex function (f(z) = u(x, y) + iv(x, y)) must satisfy in order to be differentiable. The equations are (\frac{\partial u}{\partial x} = \frac{\partial v}{\partial y}) and (\frac{\partial u}{\partial y} = -\frac{\partial v}{\partial x}).

An analytic function is a complex function that is holomorphic at every point in its domain. A holomorphic function is a function that has a derivative at every point in its domain.

The residue of a function (f(z)) at a point (z = a) is given by (\lim_{z\to a} (z-a)f(z)). In this case, (\lim_{z\to 0} (z-0)f(z) = \lim_{z\to 0} z\frac{1}{z^2} = \lim_{z\to 0} \frac{1}{z} = \infty). Therefore, the residue of (f(z) = \frac{1}{z^2}) at (z = 0) is ($\infty$).

The value of the integral (\int_C \frac{1}{z} dz) is given by (2\pi i), where (C) is a positively oriented simple closed curve around the origin. This is known as Cauchy's integral theorem.

The residue theorem is a powerful tool for evaluating integrals of complex functions around closed curves. It states that the value of the integral is equal to (2\pi i) times the sum of the residues of the function at its poles inside the curve.