Complex functions
Objective. Analyse limits and continuity in the complex plane.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Complex functions f: ℂ → ℂ have limits and continuity defined exactly as in real analysis but using the complex modulus: lim(z→a) f(z) = L means for every ε > 0 there is δ > 0 with |z−a| < δ implying |f(z)−L| < ε. Unlike real limits approached from two directions, complex limits must agree along every possible path approaching a, making complex continuity a stronger condition to verify. For f(z) = z̄/z (z ≠ 0), approaching 0 along the real axis gives limit 1, while approaching along the imaginary axis gives limit −1, so the limit at 0 does not exist.
Worked example
Determine whether the complex function f(z) = Re(z)/|z| has a limit as z → 0, considering different functions paths.
- 1Approach along the real axis with z = x, x→0⁺: f(z) = x/|x| = 1 for x > 0.
- 2Approach along the imaginary axis with z = iy: f(z) = 0/|iy| = 0 for all y ≠ 0.
- 3The two approach paths give different limiting values, 1 and 0 respectively.
- 4Since limits differ by path, the limit of f(z) as z→0 does not exist.
Answer: The limit does not exist
Concept mastery
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Practice
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Start practisingCommon mistake. Students check only one approach path, such as along the real axis, and wrongly conclude a complex limit exists; genuine existence requires checking that every path gives the same limiting value.
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