**Exercise 1. **Solve Exercise 44 in Chapter 1 of the book.

**Exercise 2.** ** **Solve Exercise 3 in Chapter 1 of the book.

**Exercise 3. ** Solve Exercise 39 in Chapter 1 of the book.

**Exercise 4.*** *The heat kernel on is given by .

- By using the subordination identity show that for ,
- The Bernoulli numbers are defined via the series expansion By using the previous identity show that for , ,

**Exercise 5.*** *Show that the heat kernel on the torus is given by