Daily Assignment for Monday, April 7

Due Wednesday, April 9 on Canvas.

• In class, I sketched some properties of the \( \Gamma \)-function. Fill in the details in the problems below. We are using the definition \( \Gamma(s) = \int_0^\infty t^{s-1} e^{-t} dt \).
• Show that the integral defining \( \Gamma(s) \) is absolutely convergent for \( \mathrm{Re}(s) \gt 0 \). Hint : using \( |t^{s-1}| = t^{\mathrm{Re}(s)-1} \), you just have to show that \( \Gamma(s) \) is finite when \( s \in \mathbb R \). Show also that \( \int_0^\infty \log(t) t^{s-1} e^{-1} dt \) is absolutely convergent for \( \mathrm{Re}(s) \gt 0 \) for use in the next problem.
• By differentiating under the integral sign, we expect that \( \Gamma'(s) = \int_0^\infty \log(t) t^{s-1} e^{-t} dt \). Show that this expectation is correct by applying the definition of the derivative to \( \Gamma(s) \). Hint : by definition of the derivative of \( t^{s-1} \), for every \( \varepsilon \gt 0 \), the difference \( t^x - 1 - \log(t) x \) is smaller than \( x \varepsilon \) for all sufficiently small \( x \).
• Conclude that \( \Gamma \) is meromorphic at all \( s \in \mathbb C \).
• Show that \( \Gamma \) never takes the value \( 0 \). Warning : this one is pretty hard.
• Make sure you know the statement of Liouville's theorem. It is in §7.7.2 of Needham's book, but you may prefer the proof from exercise 2 on p. 506 of Needham or from §7.2 of Howie. Optional : read all of §7.2 of Howie.
• Assign yourself a grade for this assignment :
  1. At least 60 minutes.
  2. About 45 minutes.
  3. About 30 minutes.
  4. About 15 minutes.
  Submit your grade as a submission comment with your assignment. This comment must be the first comment and it must be exactly 1 character long or the grade won't be registered.