What does it mean when $|z| = 2$ is the curve in a contour integral?

Question

B) Evaluate:

$$\oint_{|z|=2} \tan{z}\,dz$$

Specifically looking at B on the image. What is meant by $|z|=2$?

Answer 1

$|z| = 2$ is the circle of radius $2$ centered at the origin. Typically, when one writes an integral like \begin{align} \oint_{|z| = r} f(z) \, dz, \end{align} what is meant is that we have to consider the path $\gamma: [0,2\pi] \to \Bbb{C}$ given as $\gamma(t) := re^{it}$ (so the orientation of the path is counter clockwise). And then, really we're supposed to compute a line integral: \begin{align} \int_{\gamma} f(z) \, dz &= \int_0^{2\pi} f(\gamma(t)) \cdot \gamma'(t) \, dt \\ &= \int_0^{2\pi} f(r e^{it}) \cdot ir e^{it} \, dt. \end{align}

Answer 2

It means the circle of radius $2$ centred at $0 + 0i = 0$.

Answer 3

It is the boundary in the complex plane where the magnitude of the complex number is 2. For example, let $z = a+bi$ be a complex number that lies on the complex plane, the $a,b$ must satisfy $a^2+b^2 =z^2$, it very similar to a circle equation with radius 2, only with complex numbers.

Answer 4

If z = x + iy then |z| is given by $\sqrt( x^2 + y^2)$. Therefore |z| = 2 represents $ x^2+ y^2 =4$, the boundary of the circle of radius 2 passing through the origin.