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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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**Problem Statement** Determine the order of the zero at \( z = 0 \) for the following functions: 1. \( (e^z - 1)^2 \) 2. \( \sin z \sin 2z \sin 3z \) **Explanation** In this mathematical problem, we are tasked with finding the order of the zeros at the point \( z = 0 \) for two given functions. The order of a zero refers to the number of times the function approaches zero as \( z \) approaches the specified point. * For the function \( (e^z - 1)^2 \), we analyze the behavior of the exponential function minus one, raised to the power of two, near \( z = 0 \). * The second function is a product of sine functions: \( \sin z \), \( \sin 2z \), and \( \sin 3z \). The order at zero will be influenced by the zeros of each individual sine function at that point.
Transcribed Image Text:**Problem Statement** Determine the order of the zero at \( z = 0 \) for the following functions: 1. \( (e^z - 1)^2 \) 2. \( \sin z \sin 2z \sin 3z \) **Explanation** In this mathematical problem, we are tasked with finding the order of the zeros at the point \( z = 0 \) for two given functions. The order of a zero refers to the number of times the function approaches zero as \( z \) approaches the specified point. * For the function \( (e^z - 1)^2 \), we analyze the behavior of the exponential function minus one, raised to the power of two, near \( z = 0 \). * The second function is a product of sine functions: \( \sin z \), \( \sin 2z \), and \( \sin 3z \). The order at zero will be influenced by the zeros of each individual sine function at that point.
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