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<!doctype html>
<html>
<head>
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<title>Complex Analysis</title>
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<meta name="dc.title" CONTENT="Complex Analysis">
<meta name="dc.creator" CONTENT="Juan Carlos Ponce Campuzano">
<meta name="dc.date" CONTENT="26-01-2019">
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</head>
<body>
<div id="wrapper">
<header><p><span id="prev"><a href="domain_coloring.html">←</a></span><span id="title">
<a href="table_of_contents.html">Complex Analysis</a></span><span id="next">
<a href="curves_in_the_complex_plane.html">→</a></span><div class="clear"></div></p></header>
<h1>The Complex Power Function</h1>
<hr>
<div class="w3-responsive">
<p>The <em>generalized complex power function</em> is defined as:
\begin{eqnarray}\label{gcp}
f(z) = z^c = \exp(c\log z), \quad \text{with}\quad z\neq 0.
\end{eqnarray}
</p>
<div class="w3-responsive">
<p>Due to the multi-valued nature of $\log z$, it follows that (\ref{gcp}) is also multi-valued for any non-integer value of $c$, with a branch point at $z=0$. In other words
\begin{eqnarray*}
f(z) = z^c&= &\exp\left(c \log z\right) = \exp\left[c \left( \text{Log}\,z + 2n\pi i \right)\right], \text{ with } n\in \mathbb Z.
%&= & \exp\left(c \,\text{Log}\,z\right)\cdot \exp\left(2 n \pi c\, i\right) \\
%&= & \exp\left[ c \left( \ln |z| +i\, \textbf{Arg} \left(z\right)\right) \right] \cdot \exp\left(2 n \pi c\, i\right)\\
%&= & |z|^c \cdot \exp\left[ c\left(\textbf{Arg} (z) + 2n\pi \right) i\right], \quad\text{ with } n\in \mathbb Z.
\end{eqnarray*}
</p>
<p>On the other hand, we have that the <em>generalized exponential function</em>, for $c \neq 0 $, is defined as:
\begin{eqnarray}\label{gef}
f(z)=c^z=\exp(z\log c)=\exp\left[z \left(\text{Log}\,c +2 n \pi \, i\right)\right],
\end{eqnarray}
with $n\in \mathbb Z$.</p>
<p>
Notice that (\ref{gef}) possesses no branch point (or any other type of singularity) in the infinite complex $z$-plane. Thus, we can regard the equation (\ref{gef}) as defining a set of independent single-value functions for each value of $n$.</p>
</div>
<p>This is the reason why the multi-valued <em>nature</em> of the function $f(z)=z^c$ differs from the multi-valued function $f(z)=c^z$.</p>
<div class="w3-responsive">
<p>Typically, the $n=0$ case is the most useful, in which case, we would simply define:
$$w=c^z=\exp(z\log c)=\exp(z\,\text{Log}\,c),$$
with $c\neq 0$.</p>
</div>
<p>This conforms with the definition of exponential function
$$e^z=e^x(\cos y +i\sin y )$$
where $c = e$ (the Euler constant).</p>
<p>Use the following applet to explore functions (\ref{gcp}) and (\ref{gef}) defined on the region $[-3,3]\times[-3,3]$. The enhanced phase portrait is used with contour lines of modulus and phase. Drag the points to change the value of $c$ in each case. You can also deactivate the contour lines, if you want.</p>
<!--
<div class="w3-hide-small w3-center">
<div class="w3-row-padding">
<div id="thomas" class="w3-half w3-container w3-margin-bottom">
<figure class="w3-center">
<iframe name="z^c" src="../applets/p5js/powerfunction/zpowc/index.html" width="350px" height="350px" style="border:none;" ></iframe>
<figcaption>
$f(z)=z^c$.
</figcaption>
</figure>
</div>
<div id="aizawa" class="w3-half w3-container w3-margin-bottom">
<figure class="w3-center">
<iframe name="c^z" src="../applets/p5js/powerfunction/cpowz/index.html" width="350px" height="350px" style="border:none;" ></iframe>
<figcaption>
$f(z)=c^z$.
</figcaption>
</figure>
</div>
</div>
</div><!--responsive ends-->
<!--
<div class="w3-hide-large w3-hide-medium">
<p class="callout"><strong>Oops!</strong> Sorry, the applet is not supported for small screens.</p>
</div>
-->
<div class="w3-hide-small w3-hide-medium">
<div id="applet">
<script id="csinit" type="text/x-cindyscript">
grid1=true;
grid2=true;
</script>
<script id="csdraw" type="text/x-cindyscript">
N.y = -2.5;
n = round(5*(N.x+4));
//n = min(-17)
n = max(-22, n);
N.x = n/5-4;
hsvToRGB(h, s, v) := (
regional(j, p, q, t, f);
h = (h-floor(h))*6;
j = floor(h);
f = h - j;
p = 1 - s;
q = 1 - s*f;
t = 1 - s*(1-f);
if(j == 0, [1, t, p],
if(j == 1, [q, 1, p],
if(j == 2, [p, 1, t],
if(j == 3, [p, q, 1],
if(j == 4, [t, p, 1],
if(j == 5, [1, p, q]))))))*v
);
color(z) := ( //what color should be given to a complex number z?
regional(n0, grey1, grey2);
n0 = 12;
z = log(z)/2/pi;// + i * N.x * pi;
zfract = n0*z - floor(n0*z); //value of n*z in C mod Z[i]
//grey1 = im(zfract);
//grey2 = re(zfract);
grey1 = if(grid1, im(zfract), 1);
grey2 = if(grid2, re(zfract), 1);
hsvToRGB(im(z), 1.0, .6+.5*re(sqrt(grey1 * grey2)))
);
t(z) := (A.x + i*A.y)^z;
u(z) := z^(B.x + 6 + i * B.y);
colorplot(
//z = zoom*complex(#);
z = complex(#);
color(t(z));
,
(-3,-3),(3,3)
);
colorplot(
z = (complex(#)+6);
color(u(z));
,
(-9,-3),(-3,3)
);
shape1 = polygon([(-9,2.5), (-6,2.5), (-6,3), (-9,3)]);
fill(shape1,color->(0.9,0.9,0.9));
draw(shape1, color->(0.9,0.9,0.9));
shape2 = polygon([(-3,2.5), (0,2.5), (0,3), (-3,3)]);
fill(shape2,color->(0.9,0.9,0.9));
draw(shape2, color->(0.9,0.9,0.9));
drawtext((-2.9,2.7), "$c^z$, $c = (" + round(A.x * 100)/100 + " , " + round(A.y * 100)/100 + ")$", color->(0,0,0));
drawtext((-8.9,2.7), "$z^c$, $c = (" + round((B.x + 6 ) * 100)/100 + " , " + round(B.y * 100)/100 + ")$", color->(0,0,0));
linesize(3);
draw((-3,-3),(-3,3), color->[0,0,0]);
drawtext((A.x+0.06,A.y+0.06),"$c$", color->(0.98,0.98,0.98));
drawtext((B.x+0.06,B.y+0.06),"$c$", color->(0.98,0.98,0.98));
if(A.x < -3, A.x=-3);
if(A.y < -3, A.y=-3);
if(A.x > 3, A.x=3);
if(A.y > 3, A.y=3);
if(B.x < -9, B.x=-9);
if(B.y < -3, B.y=-3);
if(B.x > -3, B.x=-3);
if(B.y > 3, B.y=3);
//draw((-8.5,-2.5),(2.5,-2.5), color->[0,0,0]);
//drawtext((N.x-0.2,N.y+0.3),"$n = " + n + "$", color->(0.98,0.98,0.98));
</script>
<div id="CSCanvas" style="border:2px solid black"></div>
<br>
<input id="ch1" onclick="check1()" checked type="checkbox" >Phase
<input id="ch2" onclick="check2()" checked type="checkbox" >Modulus
<script type="text/javascript">
var cdy = CindyJS({canvasname:"CSCanvas",
scripts: "cs*",
geometry: [
{name:"A", type:"Free", pos:[1.2, 0.8], color:[0,0.9,1]},
{name:"B", type:"Free", pos:[-4.5, 0.6], color:[0,0.9,1]},
//{name:"N", type:"Free", pos:[-4, -2], color:[0.9,0.9,1], size:8}
],
animation: {autoplay: true},
ports: [{
id: "CSCanvas",
width: 800,
height: 400,
transform: [ { visibleRect: [-9,-3, 3, 3] } ]
}],
use: ["CindyGL", "katex"]
});
var check1=function(){
if(document.getElementById("ch1").checked){
cdy.evokeCS('grid1=true;');
} else {
cdy.evokeCS('grid1=false;');
}
}
var check2=function(){
if(document.getElementById("ch2").checked){
cdy.evokeCS('grid2=true;');
} else {
cdy.evokeCS('grid2=false;');
}
}
</script>
</div><!--ends applet-->
</div><!--end hidesmall-->
<div class="w3-hide-large w3-khaki w3-round w3-padding w3-panel">
<p><i class="fas fa-sad-tear"></i> Sorry, the applet is not supported for small screens. Rotate your device to landscape. Or resize your window so it's more wide than tall.</p>
</div>
<br>
<hr>
<p><strong>Final remark: </strong>In practice, many textbooks treat the <em>generalized exponential function</em> as a single-valued function, $c^z=\exp(z\,\text{Log } c )$, only
when $c$ is a positive real number. For any other value of $c$, the multi-valued function $c^z=\exp(z \log c )$ is preferred.</p>
</p>
<br>
<!-- link to the next page -->
<a href="curves_in_the_complex_plane.html"><p class="nextPage">NEXT: Curves in the Complex Plane</p></a>
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<p>[ <a href="../">intro</a>, <a href="https://github.com/complex-analysis">source</a>, <a href="https://github.com/complex-analysis/complex-analysis.github.io/issues">issues</a> ]</p>
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</a>
<br/>
<strong style="font-size: 18px">ISBN: 978-0-6485736-0-9</strong> <br>
<strong style="font-size: 18px">
<span>© Juan Carlos Ponce Campuzano 2019-<script>document.write(new Date().getFullYear());</script></span>
</strong>
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