--- title: "Complex Functions and the Complex Plane as a Domain" description: "Treat a function of a complex variable as a map of the plane to itself, and set up limits and continuity in that setting." canonical: https://lightmysky.com/learn/mathematics/complex-functions-and-the-complex-plane-as-a-domain-mt_hMo851VTEs source: https://lightmysky.com/learn/mathematics/complex-functions-and-the-complex-plane-as-a-domain-mt_hMo851VTEs.md retrieved: 2026-11-12 --- > **Agent view.** This is the Markdown twin of the page, for tools and assistants. > When to use this site, and the call that answers each job: https://lightmysky.com/agent-instructions.md > API description (OpenAPI 3.1): https://lightmysky.com/openapi.json · Authentication: https://lightmysky.com/auth.md > Pricing: https://lightmysky.com/pricing.md · Catalog: https://lightmysky.com/llms.txt · Full catalog: https://lightmysky.com/llms-full.txt > Every machine-readable file on this domain: https://lightmysky.com/.well-known/ai-catalog.json > Ask for Markdown with `Accept: text/markdown`, a `.md` address, or `?mode=agent`. # Complex Functions and the Complex Plane as a Domain Treat a function of a complex variable as a map of the plane to itself, and set up limits and continuity in that setting. Subject: Mathematics · Area: Complex Analysis · Ages 19 to 20 Page: https://lightmysky.com/learn/mathematics/complex-functions-and-the-complex-plane-as-a-domain-mt_hMo851VTEs ## Ready when they can - Describe the image of a line or circle under a simple complex map - Define a limit of a complex function over its domain, explain why the epsilon-delta condition controls every sufficiently near input, and state what continuity at a point demands - Split a complex function into its real and imaginary parts as two real functions of two variables ## Lesson: Functions that remap the whole plane A complex function f takes a domain D, a set of points in the plane, and sends each z in D to one output f(z). Write z = x + iy and f(z) = u(x,y) + iv(x,y). The real part is u and the imaginary part is v, not iv. Both u and v are real functions of the two inputs x and y. **Example.** Split w = z² by expansion. With z = x + iy, z² = (x + iy)² = x² + 2ixy + (iy)² = x² - y² + 2ixy, using i² = -1. So u(x,y) = x² - y² and v(x,y) = 2xy. Check: at (1,2), u = 1 - 4 = -3 and v = 2·1·2 = 4. **Example.** For example, under w = z + i the real axis z = t maps to w = t + i, and every point with imaginary part 1 is attained as t runs over the reals. To map a whole set, do not sample a few points. Parametrize the input with one real parameter t, compute the image, then show every point of the proposed image is attained. Under w = 2z + i, the circle |z| = 1 maps to |w - i| = 2, and z = (w - i)/2 shows full attainment. Under w = z², the line z = 1 + it gives w = 1 - t² + 2it, so u = 1 - t² and v = 2t, giving the parabola v² = 4(1 - u). Every point is attained since t = v/2 recovers the parameter. An accumulation point of D has points of D other than itself arbitrarily nearby. A complex limit controls every near input. The limit of f(z) as z tends to z0 is L when for every epsilon > 0 there is delta > 0 such that 0 < |z - z0| < delta implies |f(z) - L| < epsilon, for z in D with z0 an accumulation point of D. **Example.** Polynomials are continuous: the limit of z^2 + 1 at any point equals its value there. For a punctured quotient, z^2 - a^2 = (z - a)(z + a), so (z^2 - a^2)/(z - a) = z + a for z ≠ a, and the limit as z tends to a is 2a. **Example.** Two conflicting paths kill a limit. For f(z) = xy/(x² + y²) with z = x + iy, along the real axis (y = 0) the value is 0, but along the diagonal y = x the value is x²/(2x²) = 1/2. The two paths disagree, so the limit at 0 does not exist. A few agreeing paths cannot prove a limit. For an all-path bound, |z|² tends to 0 because |z|² - 0 = |z|² < epsilon whenever |z| < sqrt(epsilon), so the limit is 0 from every direction at once. Continuity at z0 needs f(z0) defined and equal to the limit. **Recap.** Split maps into u and v, prove whole images with parameters or moduli, and demand one limit that survives every nearby input. ## Practice 18 questions on this page, each with its working shown. ## Needs first - [The Imaginary Unit and Complex Arithmetic](https://lightmysky.com/learn/mathematics/the-imaginary-unit-and-complex-arithmetic-mt_Faemafn550) - [Functions of Several Variables and Level Curves](https://lightmysky.com/learn/mathematics/functions-of-several-variables-and-level-curves-mt_u04fZ_XiJ8) - [The Argand Plane, Modulus and Argument](https://lightmysky.com/learn/mathematics/the-argand-plane-modulus-and-argument-mt_Wq01c5UN9e) ## Opens up - [Complex Differentiability and the Cauchy-Riemann Equations](https://lightmysky.com/learn/mathematics/complex-differentiability-and-the-cauchy-riemann-equations-mt_nZNmZcl9Ss)