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Find the order of the zero at z = 0 of the functions (e² − 1)² and sin z sin 2z sin 3z.

Find the order of the zero at z = 0 of the functions (e² − 1)² and sin z sin 2z sin 3z.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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**Find the order of the zero at \( z = 0 \) of the functions \( (e^z - 1)^2 \) and \( \sin z \sin 2z \sin 3z \).** To determine the order of the zero at \( z = 0 \) for each function, we analyze the behavior of the functions as \( z \) approaches zero. 1. **Function \( (e^z - 1)^2 \):** - The function \( e^z \) can be expanded using its Taylor series as \( e^z = 1 + z + \frac{z^2}{2} + \cdots \). - Therefore, \( e^z - 1 = z + \frac{z^2}{2} + \cdots \). - The term \( (e^z - 1) \) has a leading order of \( z \) (first-order) in its expansion. - Thus, \( (e^z - 1)^2 = (z + \text{higher order terms})^2 = z^2 + \text{higher order terms} \). - Therefore, the order of the zero of \( (e^z - 1)^2 \) at \( z = 0 \) is 2. 2. **Function \( \sin z \sin 2z \sin 3z \):** - For small values of \( z \), the Taylor series expansion for \( \sin z \) is \( z - \frac{z^3}{6} + \cdots \). - Thus, \( \sin z = z \), \( \sin 2z = 2z \), \( \sin 3z = 3z \) to the first order. - Therefore, \( \sin z \sin 2z \sin 3z = z \cdot 2z \cdot 3z = 6z^3 + \text{higher order terms} \). - Therefore, the order of the zero of \( \sin z \sin 2z \sin 3z \) at \( z = 0 \) is 3.
Transcribed Image Text:**Find the order of the zero at \( z = 0 \) of the functions \( (e^z - 1)^2 \) and \( \sin z \sin 2z \sin 3z \).** To determine the order of the zero at \( z = 0 \) for each function, we analyze the behavior of the functions as \( z \) approaches zero. 1. **Function \( (e^z - 1)^2 \):** - The function \( e^z \) can be expanded using its Taylor series as \( e^z = 1 + z + \frac{z^2}{2} + \cdots \). - Therefore, \( e^z - 1 = z + \frac{z^2}{2} + \cdots \). - The term \( (e^z - 1) \) has a leading order of \( z \) (first-order) in its expansion. - Thus, \( (e^z - 1)^2 = (z + \text{higher order terms})^2 = z^2 + \text{higher order terms} \). - Therefore, the order of the zero of \( (e^z - 1)^2 \) at \( z = 0 \) is 2. 2. **Function \( \sin z \sin 2z \sin 3z \):** - For small values of \( z \), the Taylor series expansion for \( \sin z \) is \( z - \frac{z^3}{6} + \cdots \). - Thus, \( \sin z = z \), \( \sin 2z = 2z \), \( \sin 3z = 3z \) to the first order. - Therefore, \( \sin z \sin 2z \sin 3z = z \cdot 2z \cdot 3z = 6z^3 + \text{higher order terms} \). - Therefore, the order of the zero of \( \sin z \sin 2z \sin 3z \) at \( z = 0 \) is 3.
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