Section 21-22: Cauchy-Riemann equations

Question 1

Definition of CR
aka theorem page 64

Suppose that $f(z) = u(x,y) +iv(x,y)$ and f is differentiable at $z_0 = u_0 + i v_0$ then…

Answer 1

1. the first order partial derivatives of u and v exists at $(x_0,y_0)$ and
2. satisfy the CR equations
   $$u_x = u_y, \quad u_y = -v_x$$
   at $(x_0,y_0)$ also
3. $$f’(x_0,y_0) = u_x(x_0,y_0) + i v_x(x_0,y_0)$$

Question 2

Theorem page 66
Conditions for $f’(z_0)$ to exsist

Same thm pg 64, but this tells us sufficient conditions for differentiability at that pt

(CR only says what deriv is if given that f(z) is diff at pt $z+0$)

Answer 2

1. the $\mathbf{first order}$ partial derivatives $u_x, u_y, v_x, v_y$ $\mathbf{exists}$ everywhere in the neighbourhood
2. these partial derivs are $\mathbf{continuous}$ and satisfy the CR equations at at $(x_0,y_0)$

then $f’(z_0)$ exists and
$$f’(x_0,y_0) = u_x(x_0,y_0) + i v_x(x_0,y_0)$$

Question 3

Formal definition of diifferentuability of multivariable function $u = u(x,y)$ at pt $(x_0,y_0)$

Answer 3

u is differentiable at pt if
1. the first order partial derivatives of u exist in a neighbourhood of $(x_0,y_0)$ and

2. are continuous at $(x_0,y_0)$

then u is differentiable at $(x_0,y_0)$

Question 4

CR equations in polar coordinates

let $f(z) = u(r,\theta) +i v(r,\theta)$ be defined in an $\epsilon$ neighbourhood of a non-zero point $z_0 = r_0 e^{i\theta_0}$ and suppose that

Answer 4

1. the first order partial derivatives $u_r,u_{\theta},v_r,v_{\theta} \mathbf{exist everywhere in the neighbourhood}$
2. these partials are also $\mathbf{continous}$ at $(r_0,\theta_0)$ and
3. Satisfy the CR equations in polar form at $(r_0,\theta_0)$ :
   $$ru_r = v_{\theta}, \quad u_{\theta} = -r v_r$$

then the dervative at z_0 exists and
$$f’(z_0) = e^{-i\theta_0}(u_r(r_0,\theta_0) + i v_r(r_0,\theta_0)$$