Standard Functions

Reading

Section 3.2

Problems

Practice problems:

(Pages 41-42) 12, 13, 14, 15, 17, 18

Challenge: 16

Be ready to present problems 14, 15

Topics to know

1. Can define \(e^z\) via its power series. We will offer an alternative definition.
   • Denote the function we are after as \(f(z)\). We are looking for:
     • \(f\) analytic.
     • \(f(z+w) = f(z)f(w)\).
     • \(f\) is the exponential when restricted to real numbers.
   • If \(z=x+iy\), then \(f(z) = f(x)f(iy)\). But \(f(x)= e^x\). Suppose \(f(iy) = A(y) + i B(y)\), where \(A,B\) are differentiable real functions.
   • Then \(f(z) = e^x A(y) + i e^x B(y)\). Also \(B(0) = 0\) and \(A(0) = 1\).
   • Cauchy-Riemann equations must hold. We compute \(f_x(z) = f(z)\), and \(f_y(z) = e^x A'(y) + i e^x B'(y)\). In order for \(f_y = if_x\) to hold, must have \(B' = A\) and \(A' = -B\).
   • These two real-variable differentiable equations are solved by \(A(y) = \cos y\) and \(B(y) = \sin y\).
   • We therefore have \(f(z) = e^x (\cos y + i\sin y) = e^x \textrm{cis} y\).
2. Properties:
   • \(|e^z| = e^{\textrm{Re}(z)}\).
   • \(e^z\neq 0\).
   • \(e^{iy} = \textrm{cis} y\).
   • \(e^z = a\) has infinitely many solutions (explain!!!!).
   • \((e^z)' = e^z\). Check it!
3. Can define:
   • \(\sin z = \frac{1}{2i}\left(e^{iz} - e^{-iz}\right)\).
   • \(\cos z = \frac{1}{2}\left(e^{iz} + e^{-iz}\right)\).
   • These make sense: Can use the two equations \(e^{\pm iy} = \cos y \pm i\sin y\) to show that when \(z\) is real these equations are indeed correct.