3.4 Analytic Functions

$\newenvironment {prompt}{}{} \newcommand {\ungraded }[0]{} \newcommand {\ffrac }[2]{\frac {\text {\footnotesize $#1$}}{\text {\footnotesize $#2$}}} \newcommand {\npnoround }[0]{\nprounddigits {-1}} \newcommand {\npnoroundexp }[0]{\nproundexpdigits {-1}} \newcommand {\npunitcommand }[1]{\ensuremath {\mathrm {#1}}} \newcommand {\RR }[0]{\mathbb R} \newcommand {\R }[0]{\mathbb R} \newcommand {\C }[0]{\mathbb C} \newcommand {\N }[0]{\mathbb N} \newcommand {\Z }[0]{\mathbb Z} \newcommand {\dis }[0]{\displaystyle } \newcommand {\d }[0]{\mathop {}\!d} \newcommand {\l }[0]{\ell } \newcommand {\ddx }[0]{\frac {d}{\d x}} \newcommand {\ppx }[0]{\frac {\partial }{\partial x}} \newcommand {\ppy }[0]{\frac {\partial }{\partial y}} \newcommand {\zeroOverZero }[0]{\ensuremath {\boldsymbol {\tfrac {0}{0}}}} \newcommand {\inftyOverInfty }[0]{\ensuremath {\boldsymbol {\tfrac {\infty }{\infty }}}} \newcommand {\zeroOverInfty }[0]{\ensuremath {\boldsymbol {\tfrac {0}{\infty }}}} \newcommand {\zeroTimesInfty }[0]{\ensuremath {\small \boldsymbol {0\cdot \infty }}} \newcommand {\inftyMinusInfty }[0]{\ensuremath {\small \boldsymbol {\infty -\infty }}} \newcommand {\oneToInfty }[0]{\ensuremath {\boldsymbol {1^\infty }}} \newcommand {\zeroToZero }[0]{\ensuremath {\boldsymbol {0^0}}} \newcommand {\inftyToZero }[0]{\ensuremath {\boldsymbol {\infty ^0}}} \newcommand {\numOverZero }[0]{\ensuremath {\boldsymbol {\tfrac {\#}{0}}}} \newcommand {\dfn }[0]{\textbf } \newcommand {\unit }[0]{\mathop {}\!\mathrm } \newcommand {\eval }[1]{ #1 \bigg |} \newcommand {\seq }[1]{\left ( #1 \right )} \newcommand {\epsilon }[0]{\varepsilon } \newcommand {\iff }[0]{\Leftrightarrow } \DeclareMathOperator {\arccot }{arccot} \DeclareMathOperator {\arcsec }{arcsec} \DeclareMathOperator {\arccsc }{arccsc} \DeclareMathOperator {\si }{Si} \DeclareMathOperator {\proj }{proj} \DeclareMathOperator {\scal }{scal} \DeclareMathOperator {\cis }{cis} \DeclareMathOperator {\Arg }{Arg} \DeclareMathOperator {\Rep }{Re} \DeclareMathOperator {\Imp }{Im} \DeclareMathOperator {\sech }{sech} \DeclareMathOperator {\csch }{csch} \DeclareMathOperator {\Log }{Log} \newcommand {\tightoverset }[2]{\mathop {#2}\limits ^{\vbox to -.5ex{\kern -0.75ex\hbox {$#1$}\vss }}} \newcommand {\arrowvec }[0]{\overrightarrow } \newcommand {\vec }[0]{\mathbf } \newcommand {\veci }[0]{{\boldsymbol {\hat {\imath }}}} \newcommand {\vecj }[0]{{\boldsymbol {\hat {\jmath }}}} \newcommand {\veck }[0]{{\boldsymbol {\hat {k}}}} \newcommand {\vecl }[0]{\boldsymbol {\l }} \newcommand {\utan }[0]{\vec {\hat {t}}} \newcommand {\unormal }[0]{\vec {\hat {n}}} \newcommand {\ubinormal }[0]{\vec {\hat {b}}} \newcommand {\dotp }[0]{\bullet } \newcommand {\cross }[0]{\boldsymbol \times } \newcommand {\grad }[0]{\boldsymbol \nabla } \newcommand {\divergence }[0]{\grad \dotp } \newcommand {\curl }[0]{\grad \cross } \newcommand {\vector }[1]{\left \langle #1\right \rangle }$

We determine if a function is analytic using the Cauchy-Riemann equations.

We say that $f(z)$ is analytic at $z_0$ if $f$ is differentiable on $D(z_0, r)$ for some $r>0$ . Furthermore, $f(z)$ is called analytic on a region $R$ if it is analytic at each point in $R$ .

example 1 Recall from section 3.2 that the function $f(z) = \overline {z}^2$ is differentiable at $z=0$ and nowhere else. Thus, $f(z) = \overline {z}^2$ is not differentiable on any disk of positive radius. Hence, $f(z) = \overline {z}^2$ is nowhere analytic.

(problem 1) Where is the function $f(z) = (x^3 + 3xy^2) +i(3x^2y+y^3)$ analytic?
$u_x = \answer {3x^2+3y^2} \quad \text {and} \quad u_y = \answer {6xy}$
$v_x = \answer {6xy} \quad \text {and} \quad v_y = \answer {3x^2 + 3y^2}$
Complete the sentence: $f$ is differentiable

on the real axis only on the imaginary axis only on both the real and imaginary axes at the origin only nowhere

Complete the sentence: $f$ is analytic

on the real axis only on the imaginary axis only on both the real and imaginary axes at the origin only nowhere

Here is a video solution to problem 1:

example 2 The function $f(z) = e^z$ is differentiable at every point in $\C$ and hence it is analytic everywhere in $\C$ .

(problem 2a) Where is the function $f(z) = \cos (z)$ analytic?

on punctured plane on the slit plane on entire plane at the origin only nowhere

(problem 2b) Where is the function $f(z) = \Log (z)$ analytic?

on punctured plane on the slit plane on entire plane at the origin only nowhere

(problem 2c) Where is the function $f(z) = \frac {1}{z}$ analytic?

on punctured plane on the slit plane on entire plane at the origin only nowhere

Like the real variable case, an analytic function whose derivative is zero is a constant.

Suppose $f(z)$ is analytic in a disk, $D$ . If $f'(z) =0$ on $D$ , then $f(z)$ is constant on $D$ .

Proof: First note that since $f'=0$ on $D$ , the partial derivatives $u_x, u_y, v_x$ and $v_y$ are all zero on the disk. Suppose $z_1, z_2 \in D$ with $\Imp z_1 = \Imp z_2$ . Then on the horizontal segment between $z_1$ and $z_2$ , $u_x = 0$ implies that $u$ is constant on the segment and $v_x =0$ on the segment implies that $v$ is constant on the segment as well. Now suppose $z_1, z_2 \in D$ with $\Rep z_1 = \Rep z_2$ . Then on the vertical segment between $z_1$ and $z_2$ , $u_y = 0$ implies that $u$ is constant on the segment and $v_y =0$ on the segment implies that $v$ is constant on the segment as well. Finally, any two points $z_1, z_2 \in D$ can be connected by a horizontal and a vertical segment and $f$ is constant on these segments, so $f(z_1) = f(z_2)$ . Since $z_1$ and $z_2$ were arbitrary, $f(z)$ is constant on $D$ . $\blacksquare$

In a vane similar to the previous theorem, if the modulus of an analytic function is constant, then the function itself is constant.

Suppose $f(z)$ is analytic in a disk $D$ . If $|f(z)| = m$ on $D$ , then $f(z)$ is constant on $D$ .

Proof: If $m =0$ , the $f$ is identically zero, so suppose $m>0$ . The definition of modulus gives $u^2(x,y) + v^2(x,y) = m^2$ Taking the partial derivative of both sides with respect to $x$ and with respect to $y$ gives the system $2uu_x +2vv_x = 0$ $2uu_y + 2vv_y = 0$ Dividing by $2$ and using the Cauchy-Riemann equations selectively gives $uu_x -vu_y = 0$ $uu_y + vu_x = 0$ Multiplying the first equation by $u$ and the second equation by $v$ gives $u^2u^x - uvu_y=0$ $uvu_y + v^2u_x = 0$ adding these equations together gives $\left (u^2 +v^2\right )u_x = m^2u_x = 0$ which gives $u_x = 0$ on $D$ . We could have eliminated $u_x$ instead of $u_y$ which would have led to $u_y =0$ on $D$ . Hence $u$ is constant on $D$ . Moreover, by the Cauchy-Riemann equations $v_x=0$ and $v_y=0$ on $D$ and so $v$ is constant on $D$ as well. Thus $f = u+iv$ is constant on $D$ . $\blacksquare$

(problem 3a) Solve the system for $u_y$ : $uu_x -vu_y = 0$ $uu_y + vu_x = 0$

$u_y= 0$ $u_y = -u_x$ $u_y = m^2$

A function that is analytic at every point in $\C$ is called entire.

(problem 3b) Select all of the functions below which are entire.

$z^3 + 2iz^2 -5z -i$ $e^z$ $\sin z$ $\cos z$ $1/z$ $\Log z$

Here is a video solution to problem 3b:

2024-09-27 13:51:00