Proving the Limit of a Parabola Delta Function

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SUMMARY

The forum discussion centers on proving the limit of a parabola delta function defined as L = lim (k→∞) ∫[-3/(4k), 3/(4k)] f(x)[-16k³/9 x² + k] dx = f(0). Participants argue that as k approaches infinity, the parabola becomes infinitely tall and narrow, converging to a delta function. The discussion also touches on the filtering property of the Dirac delta function and the implications of using different types of functions, such as continuous and differentiable functions, in this context.

PREREQUISITES
  • Understanding of limits in calculus
  • Familiarity with the Dirac delta function and its properties
  • Knowledge of definite integrals and the Mean Value Theorem
  • Experience with mathematical software like Mathematica for verification
NEXT STEPS
  • Explore the properties of the Dirac delta function in detail
  • Learn about the Lebesgue integral and its applications in proving limits
  • Investigate the implications of using continuous versus differentiable functions in limit proofs
  • Study the Intermediate Value Theorem and its role in evaluating definite integrals
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Mathematicians, physicists, and students studying advanced calculus or functional analysis, particularly those interested in the applications of delta functions in integrals.

  • #31
Looks good to me. :smile:

As long as f is continuous, and the area beneath the curves formed by each member in your sequence equals 1, then this argument really shows that ANY such sequence is good enough to represent the delta function(al).
 

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