Prove That f1(x), f2(x) and f3(x) are Orthogonal Over (-l, l) - Engineering Mathematics 3

TL;DR

This video explains how to prove that a given set of three functions are orthogonal over a specific range by manually calculating their integrals.

Transcript

Do subscribe to Ekeeda Channel and press bell icon to get updates about latest engineering HSC and IIT JEE main and advanced videos Hi students so now we are gonna see a one more numerical where we are gonna prove that the different sets which are given are orthogonal over the given range so let's see how to prove it by using the condition of ortho... Read More

Key Insights

• 👍 The condition to prove orthogonality involves calculating the integration of pairs of functions and ensuring the result is zero.
• 😫 When the functions in a set have no relation to each other, manual calculations are required to prove orthogonality.
• 👍 The video demonstrates the step-by-step process of calculating integrals for different combinations of functions to prove orthogonality.
• 😫 All three functions in the given set are shown to be orthogonal by satisfying the condition.
• 👍 Understanding how to prove orthogonality using integration is essential for achieving success in examinations.
• 😫 Orthogonal sets of functions have various applications in mathematics and related fields.
• 🖐️ The properties of odd and even functions play a role in determining the value of certain integrals.

Questions & Answers

Q: Why is it challenging to prove orthogonality when the functions in the set have no relation to each other?

When the functions in the set are completely different and lack any relationship, the condition for orthogonality involving the product of two functions cannot be conveniently applied. Manual calculations of each combination of functions are necessary in such cases.

Q: What is the condition for proving that a set of functions is orthogonal?

The condition involves calculating the integration of pairs of different functions from the set and showing that the result is equal to zero. Additionally, the integration of a function with itself should yield a value different from zero to fulfill the condition.

Q: Are all three functions in the given set orthogonal to each other?

Yes, by manually calculating the integrals for different combinations of functions, it has been shown that all three functions in the set satisfy the condition for orthogonality over the given range.

Q: How can the knowledge of orthogonal sets of functions be used in examinations?

Understanding how to prove orthogonality using integration is crucial for solving problems related to orthogonality and orthonormality. This technique can be applied to various mathematical concepts, making it a valuable skill for scoring well in exams.

Summary & Key Takeaways

• The video discusses the condition to prove that a set of functions is orthogonal, which involves calculating the integration of pairs of functions and checking if the result is equal to zero.

• The presenter explains the challenges faced when the functions in the set have no direct relation to each other, requiring a manual approach to prove orthogonality.

• Through step-by-step calculations, the presenter demonstrates how to calculate the integrals of different combinations of the functions and shows that the resulting values satisfy the conditions for orthogonality over a given range.