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Let\(S\)be a set of complex numbers. Afunction\(f\)defined on\(S\)is a rule that assigns to each \(z\)in\(S\)a complex number\(w\). The number\(w\)is called the value of\(f\)at\(z\)and is denoted by \(f(z)\); that is,\(w=f(z)\). The set\(S\)is called the domain of definition of\(f\).
If only one value of\(w\)corresponds to each value of\(z\), we say that\(w\)is asingle-valuedfunction of\(z\)or that\(f(z)\)is single-valued. If more than one value of\(w\)corresponds to each value of\(z\), we say that\(w\)is amultiple-valuedormany-valuedfunction of\(z\).
Amultiple-valuedfunction can be considered as a collection of single-valued functions, each member of which is called abranchof the function. In general, we consider one particular member as aprincipal branchof the multiple-valued function and the value of the function corresponding to this branch as theprincipal value.
Example \(\PageIndex{1}\)
The function\(w=z^{2}\)is a single-valued function of\(z\). On the other hand, if\(w=z^{\frac{1}{2}}\),then to each value of\(z\)there are two values of\(w\). Hence, the function
\(w=z^{\frac{1}{2}}\)
is a multiple-valued (in this case two-valued) function of\(z\).
Suppose that\(w=u+iv\)is the value of a function\(f\)at\(z=x+iy\), so that
\(u+iv=f(x+iy)\)
Each of the real numbers\(u\)and\(v\)depends on the real variables\(x\)and\(y\), and it follows that\(f(z)\)can be expressed in terms of a pair of real-valued functions of the real variables\(x\)and\(y\):
\(\begin{eqnarray}\label{eq1}
f(z)= u(x,y)+iv(x,y).
\end{eqnarray}\)
If the polar coordinates\(r\)and\(θ\), instead of\(x\)and\(y\), are used, then
\(u+iv=f\left ( re^{i\theta } \right )\)
where\(w=u+iv\)and\(z= re^{i\theta }\). In this case, we write
\(\begin{eqnarray}\label{eq2}
f(z)=u\left(r, \theta\right)+iv\left(r, \theta\right).
\end{eqnarray}
\)
Example \(\PageIndex{2}\)
Example 2:If\(f\left ( z \right )=z^{2}\)then
\(f\left ( x+iy \right )=\left ( x+iy \right )^{2}=x^{2}-y^{2}+i\left ( 2xy \right )\).
Hence
\(u\left ( x,y \right )=x^{2}-y^{2}\)and\(v\left ( x,y \right )=2xy\).
When we use polar coordinates, we have
\(u\left ( r,\theta \right )=r^{2}cos2\theta\)and\(v\left ( r,\theta \right )=r^{2}sin2\theta\).
Question:What happens when in either of equations (1) and (2) the function\(v\)always has a value zero?
Examples of complex functions
Polynomial functions
For\(a_{n},a_{n-1},...,a_{0}\)complex constants we define
\(p\left ( z \right )=a_{n}z^{n}+a_{n-1}z^{n-1}+\cdots +a_{1}z+a_{0}\)
where\(a_{n}\neq 0\)and\(n\)is a positive integer called thedegreeof the polynomial\(p(z)\).
Rational functions:Ratios
\(\frac{p\left ( z \right )}{q\left ( z \right )}\)
where\(p(z)\)and\(q(z)\)are polynomials and\(q(z)≠0\).
Exponential function
Exponential function:If\(z=x+iy\), the exponential function\(e^{z}\)is defined by writing
\(e^{z}=e^{x}e^{iy}\).
Because
\(e^{iy}=cos\,y+isin\,y\),
then we have
\(e^{z}=e^{x}\left ( cos\,y+isin\,y \right )\).
Logarithmic function
In a similar fashion, the complex logarithm is a complex extension of the usual real natural (i.e., base\(e\)) logarithm. In terms of polar coordinates\(z=re^{i\theta }\), the complex logarithm has the form
\(log\,z=log\left ( re^{i\theta } \right )=log\,r+log\, e ^{i\theta } =log\,r+i\theta\).
We will explore in detail this function in the following section.
Trigonometric functions
The sine and cosine of a complex variable\(z\)are defined as follows:
\(sin\,z=\frac{e^{iz}-e^{-iz}}{2i}\) and \(cos\,z=\frac{e^{iz}+e^{-iz}}{2}\).
The other four trigonometric functions are defined in terms of the sine and cosine functions with the following relations:
\(tan\,z=\frac{sin\,z}{cos\,z}\)\(cot\,z=\frac{cos\,z}{sin\,z}\)
\(sec\,z=\frac{1}{cos\,z}\)\(csc\,z=\frac{1}{sin\,z}\).
Hyperbolic trigonometric functions
The hyperbolic sine and the hyperbolic cosine of a complex variable are defined as they are with a real variable; that is,
\(sinh\,z=\frac{e^{z}-e^{-z}}{2}\)and\(cosh\,z=\frac{e^{z}+e^{-z}}{2}\).
The other four hyperbolic functions are defined in terms of the hyperbolic sine and cosine functions with the relations:
\(tanh\,z=\frac{sinh\,z}{cosh\,z}\)\(coth\,z=\frac{cosh\,z}{sinh\,z}\)
\(sech\,z=\frac{1}{cosh\,z}\)\(csch\,z=\frac{1}{sinh\,z}\).
Explore the real and imaginary components
Use the following applet to explore the real and imaginary components of some complex functions.
INTERACTIVE GRAPH
Code
Enter the following scripts inGeoGebrato explore it yourself. Open the 3D view. The symbol # indicates comments.
#Define complex function f(z) := z + 1/z#Define componentsRe = Surface(u, v, real( f(u + ί v) ), u, -5, 5, v, -5, 5)Im = Surface(u, v, imaginary( f(u + ί v) ), u, -5, 5, v, -5, 5)