Complex Analysis Practice (Interactive)

Intuition

Functions of imaginary numbers: Complex analysis extends calculus to the complex plane, where numbers have both real and imaginary parts. It reveals beautiful connections, a holomorphic function is infinitely differentiable, and its real and imaginary parts satisfy Laplace equation.

Why it matters: Complex analysis solves real engineering problems, signal processing uses Fourier transforms (complex exponentials), control theory uses the Laplace transform, and fluid dynamics uses conformal mapping.

The key insight: Cauchy integral formula says you can reconstruct a function from its values on a boundary, this seemingly abstract result has practical consequences for computing integrals and understanding analytic continuation.

University Mathematics, Complex Analysis Practice

10 auto-graded practice problems at medium to hard difficulty. Select an answer, submit, and review the explanation.

Holomorphic Functions and Cauchy-Riemann Equations

Complex Integration and Residue Theory

Series and Conformal Mappings

Common Mistakes

Assuming holomorphic implies real-differentiable in the real sense: A function $f(z) = u + iv$ being holomorphic means it satisfies the Cauchy-Riemann equations, but this does not mean $u$ and $v$ are independent real-differentiable functions. The Cauchy-Riemann equations couple $u$ and $v$, you cannot specify $u$ and $v$ independently and expect $f$ to be holomorphic.

Forgetting to check which singularities lie inside the contour: When using the residue theorem, only poles inside the contour contribute to the integral. The function $1/[(z+1)(z-3)]$ has poles at $z = -1$ and $z = 3$, but a contour $|z| = 2$ encloses only $z = -1$. Always verify $|z_k| < R$ before including a residue in your sum.

Confusing the principal branch of the logarithm with the multi-valued function: $\log(z)$ is multi-valued, and the principal branch $\operatorname{Log}(z)$ has a branch cut along the negative real axis. When computing $\log(z^n)$, you cannot directly write $n\log(z)$, the two sides may differ by multiples of $2\pi i$. Track the argument carefully around branch cuts.

Cross-References

• Site Home: Main landing page for Mathematics notes.
• Linear Algebra: Vector spaces, matrices, and linear transformations.
• Real Analysis: Rigorous treatment of real numbers and calculus.
• Practice: Practice problems for revision.