Math Problem Statement

Find the zeroes of   2 cot z z

Solution

The function you're working with is f(z)=z2cot(z)f(z) = z^2 \cot(z), and you are looking for the zeros of this function.

To find the zeros of f(z)=z2cot(z)f(z) = z^2 \cot(z), we must find values of zz such that:

z2cot(z)=0z^2 \cot(z) = 0

Since this is a product, the function will be zero if either factor is zero:

  1. z2=0z^2 = 0
  2. cot(z)=0\cot(z) = 0

1. Solving z2=0z^2 = 0

This equation gives us the solution: z=0z = 0

2. Solving cot(z)=0\cot(z) = 0

The cotangent function cot(z)\cot(z) is zero whenever z=(2n+1)π2z = \frac{(2n+1)\pi}{2} for integers nn. These are the points where the sine function is 1 or -1 (since cot(z)=cos(z)sin(z)\cot(z) = \frac{\cos(z)}{\sin(z)}).

Thus, the solutions are: z=(2n+1)π2,nZz = \frac{(2n+1)\pi}{2}, \quad n \in \mathbb{Z}

Conclusion

The zeros of f(z)=z2cot(z)f(z) = z^2 \cot(z) are: z=0andz=(2n+1)π2,nZz = 0 \quad \text{and} \quad z = \frac{(2n+1)\pi}{2}, \quad n \in \mathbb{Z}

Would you like further details on the solution?


Here are 5 related questions:

  1. What are the properties of the cotangent function?
  2. How would the graph of f(z)=z2cot(z)f(z) = z^2 \cot(z) look near the zeros?
  3. What happens to f(z)f(z) at the points where cot(z)\cot(z) is undefined?
  4. Can this problem be extended to complex numbers zz?
  5. How do the zeros of cot(z)\cot(z) relate to the zeros of sin(z)\sin(z)?

Tip: Always check the domain of trigonometric functions, as they can introduce undefined points in the function.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Complex Analysis
Roots of Functions

Formulas

z^2 cot(z) = 0
cot(z) = cos(z) / sin(z)
Zeros of cotangent: z = (2n+1)π/2, n ∈ ℤ

Theorems

Properties of Cotangent Function
Zeros of Trigonometric Functions

Suitable Grade Level

College Level