Zero of a Function

a point where a given function f(z) vanishes. Thus the zeros of the function f(z) are the same as the roots of the equation f(z) = 0. For example, the points 0, π, – π, 2π, – 2π, … are zeros of the function sin z.

The zeros of an analytic function f(z) are isolated points. For each such zero z₀ there is a natural number k such that f (z₀) = O, f′(z₀) = 0, …, f^(k-1) (z₀) = 0, but f^k(z₀) ≠ 0; k is called the order of the zero. Thus for the zero of the function 1 – cos Φ, the order k = 2. A zero is said to be simple if k = 1, and multiple if k > 1.

单词	zero of a function
释义	Zero of a Function Zero of a Function a point where a given function f(z) vanishes. Thus the zeros of the function f(z) are the same as the roots of the equation f(z) = 0. For example, the points 0, π, – π, 2π, – 2π, … are zeros of the function sin z. The zeros of an analytic function f(z) are isolated points. For each such zero z₀ there is a natural number k such that f (z₀) = O, f′(z₀) = 0, …, f^(k-1) (z₀) = 0, but f^k(z₀) ≠ 0; k is called the order of the zero. Thus for the zero of the function 1 – cos Φ, the order k = 2. A zero is said to be simple if k = 1, and multiple if k > 1.
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