Prove that 88 B₁ (x)dx = ti+k+1 k + 1
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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This question is about the Basic theory of B-splines from the textbook Numerical Analysis: Mathematics of Scientific Computing.
![**Prove the Following Mathematical Integral:**
\[
\int_{-\infty}^{\infty} B_i^k(x) \, dx = \frac{t_{i+k+1} - t_i}{k+1}
\]
**Explanation:**
The problem statement requires the proof of an equation involving an integral of a function \( B_i^k(x) \) over the entire real line, represented by the limits from \(-\infty\) to \(\infty\). The expression on the right side is a fraction which includes the terms \( t_{i+k+1} - t_i \) in the numerator, divided by \( k+1 \). Here, \( i \) and \( k \) are likely indices related to a sequence or series.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fad0d55fe-d83b-4711-86a1-cee8ecea510f%2F7b9a59a1-5f5b-4a81-a798-c3ee7bfe4c6b%2Fbmhecq_processed.png&w=3840&q=75)
Transcribed Image Text:**Prove the Following Mathematical Integral:**
\[
\int_{-\infty}^{\infty} B_i^k(x) \, dx = \frac{t_{i+k+1} - t_i}{k+1}
\]
**Explanation:**
The problem statement requires the proof of an equation involving an integral of a function \( B_i^k(x) \) over the entire real line, represented by the limits from \(-\infty\) to \(\infty\). The expression on the right side is a fraction which includes the terms \( t_{i+k+1} - t_i \) in the numerator, divided by \( k+1 \). Here, \( i \) and \( k \) are likely indices related to a sequence or series.
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