I am studying complex analysis and I need to understand the Cauchy-Goursat theorem. The theorem states:
If $f(z)$ is analytic in a simply connected domain $D$, then for any closed contour $C$ in $D$:
$$\oint_C f(z) \, dz = 0$$
Chloe Villeneuve
Editorial@villeneuve555
MethodMath editorial account. Content may be AI-assisted and is subject to correction and editorial review.
Joined December 2025
8Questions
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I'm studying linear algebra and I need to understand how to compute the rank of a matrix. I know the definition:
The rank of a matrix is the dimension of its column space (or row space).
I am studying graph theory and I need to understand minimum spanning trees (MST). A spanning tree of a connected, undirected graph is a subgraph that is a tree and includes all vertices.
I know two algorithms:
- Kruskal's algorithm: Sort edges by weight, add edges one by one if they don't form a cycle
- Prim's algorithm: Start from a vertex, repeatedly add the smallest edge connecting the tree to a vertex outside
I'm learning abstract algebra and we just defined rings. I understand the formal definition:
A ring $(R, +, \cdot)$ is a set with two binary operations such that:
1. $(R, +)$ is an abelian group
2. $(R, \cdot)$ is associative
3. Distributive laws hold
I'm studying linear algebra and I can find the inverse of a $2 \times 2$ matrix easily using the formula $A^{-1} = \frac{1}{\det(A)} \begin{pmatrix} d & -b \\ -c & a \end{pmatrix}$.
But for $3 \times 3$ matrices, the formula involving cofactors and the adjugate is very tedious. I've heard that Gauss-Jordan elimination (row reduction) is more efficient.
In my Real Analysis class, we proved that $\mathbb{Q}$ is countable by constructing a bijection with $\mathbb{N}$. The standard proof uses the diagonal array argument:
$$
\begin{matrix}
\frac{1}{1} & \frac{1}{2} & \frac{1}{3} & \frac{1}{4} & \cdots \\
\frac{2}{1} & \frac{2}{2} & \frac{2}{3} & \frac{2}{4} & \cdots \\
\frac{3}{1} & \frac{3}{2} & \frac{3}{3} & \frac{3}{4} & \cdots \\
\frac{4}{1} & \frac{4}{2} & \frac{4}{3} & \frac{4}{4} & \cdots \\
\vdots & \vdots & \vdots & \vdots & \ddots
\end{matrix}
$$
I'm studying calculus and I need to find the area between two curves. The general formula is:
$$A = \int_a^b [f(x) - g(x)] \, dx$$
When I press the sin button on a calculator, what is actually happening inside? It's not looking up a giant table of values. I heard calculators use infinite series to approximate trig functions, but how does it know when to stop adding terms to get an accurate answer?