4) Cauchy product of series Flashcards
What is the Cauchy product (convolution) of series
What is the Convolution Theorem for Series
Explain the proof of exp(x) exp(y) = exp(x + y)
What is the natural logarithm
The function ln: (0, ∞) → R is defined as the inverse function of exp
Explain the proof of ln(ab) = ln(a) + ln(b)
- Since exp and ln are mutual inverses, a = exp(ln a) and b = exp(ln b)
- ab = exp(ln(a) + ln(b))
- Taking in of both sides, we obtain ln(ab) = ln(a) + ln(b).
What is an open neighbourhood
- Let a ∈ R. An open neighbourhood of a is a set of the form (a − δ, a + δ) for some δ > 0
- This is an open neighbourhood of a is an open interval centred at a
What does it mean to be differentiable at a
What does it mean to differentiable on an open interval
A function is differentiable on an open interval if it is differentiable at every point of that interval
What is the derivative of
a constant function
If something is differentiable at a is it continuous at a
Yes, if f is differentiable at a, then f is
continuous at a
Describe the proof of differentiable at a ⇒ continuous at a
Give a counterexample to continuous at a ⇒ differentiable at a
f(x) = |x|
Describe the sum, product and quotient rules of differentiation
Suppose that the functions f, g are differentiable at a
Describe the proof of the sum rule
Describe the proof of the product rule
Describe the proof of the quotient rule
