Limits in the Complex Plane

Complex Analysis · Axiom Academy

LESSON Limits in the Complex Plane A complex limit must agree along every path of approach — that one demand is what sets it apart from real limits. 1. The Limit Is the Value f(z) Heads Toward Write to say: as z gets close to z_0 , the output f(z) gets arbitrarily close to the complex number L . The distance "close" is measured with the modulus — |z - z_0| small forces |f(z) - L| small. f(z) within of L whenever z is within of z_0 (and ) For a continuous function — and every polynomial is one — the limit is just the value: substitute z_0 . Take f(z) = z^2 and z_0 = 2+i : Here is the defining feature of a complex limit. A real limit only needs the left and right approaches to match. A complex limit needs every approach to match — straight lines, diagonals, spirals, any curve into z_0 . If they all land on one value, the limit is that value. Let z = z_0 + t with . For f(z)=z^2 this tends to L . Let z = z_0 + it . The image again tends to the same L . Any other curve into z_0 produces the identical limiting value. One common value L for all paths is exactly what asserts. For f(z) = z^2 near z_0 = 2+i , four very different paths — real axis, imaginary axis, the line y=x , and an inward spiral — all drive f(z) to the single point L = 3+4i . Continuity guarantees this: there is nothing for the paths to disagree about. 3. When Paths Disagree, the Limit Fails

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