Derivative of a distribution
Not what you're looking for?
Let f(x) and g(x) differentiable functions on R. Evaluate the derivative, in distribution sense, of the function h(x):
h(x) = f(x) if x > 0
h(x) = g(x) if x â?¤ 0.
Purchase this Solution
Solution Summary
The expert explains a bit of distribution theory while explaining the problem. Two functions are examined.
Solution Preview
Let's denote the action of a distribution T on a test function u as <T,u>. Then the derivative of T, denoted as DT, is defined by declaring that it acts on test functions as:
<DT, u> = - <T, Du>
This definition makes it consistent wih the partial integration formula when T and DT are ordinary functions (note that, by definition, test functions tend to zero at infinity fast enough that you don't get boundary terms when doing partial integration).
In this problem the distribution is the function h which is, in general, discontinuous at x = 0. if u is a general test function, we have by definition:
<Dh,u> = - <h, Du>
Since on the right hand side h is an ordinary, albeit discontinuous, function, we can write it as an ordinary ...
Purchase this Solution
Free BrainMass Quizzes
Multiplying Complex Numbers
This is a short quiz to check your understanding of multiplication of complex numbers in rectangular form.
Probability Quiz
Some questions on probability
Know Your Linear Equations
Each question is a choice-summary multiple choice question that will present you with a linear equation and then make 4 statements about that equation. You must determine which of the 4 statements are true (if any) in regards to the equation.
Graphs and Functions
This quiz helps you easily identify a function and test your understanding of ranges, domains , function inverses and transformations.
Exponential Expressions
In this quiz, you will have a chance to practice basic terminology of exponential expressions and how to evaluate them.