The Central Idea
A complex-valued function of a complex variable has the form
\[ \boxed{f:\mathbb{C}\to\mathbb{C}}. \]If
\[ z=x+iy, \]then the output
\[ f(z) \]is also a complex number.
We may write
\[ \boxed{f(z)=u(x,y)+iv(x,y)}, \]where \(u\) and \(v\) are real-valued functions of the real variables \(x\) and \(y\).
Inputs and Outputs in the Complex Plane
In ordinary real-variable calculus, a function such as
\[ f(x)=x^2 \]maps points on the real line to other points on the real line.
In complex analysis, a function such as
\[ f(z)=z^2 \]maps points in the complex plane to other points in the complex plane.
Thus a complex function may be interpreted as a transformation
\[ \boxed{z\mapsto f(z)}. \]The Complex Variable
A complex input is usually written
\[ z=x+iy, \]where
\[ x=\operatorname{Re}(z) \]and
\[ y=\operatorname{Im}(z). \]This means that a single complex variable contains two real coordinates.
Real and Imaginary Component Functions
If
\[ f(z)=u(x,y)+iv(x,y), \]then
\[ \boxed{u(x,y)=\operatorname{Re}(f(z))} \]and
\[ \boxed{v(x,y)=\operatorname{Im}(f(z))}. \]These two real-valued functions describe the complex output coordinate by coordinate.
Example: The Function \(f(z)=z^2\)
Let
\[ z=x+iy. \]Then
\[ f(z)=z^2=(x+iy)^2. \]Expand:
\[ x^2+2ixy+i^2y^2. \]Since \(i^2=-1\),
\[ f(z)=x^2-y^2+i(2xy). \]Therefore,
\[ \boxed{u(x,y)=x^2-y^2} \]and
\[ \boxed{v(x,y)=2xy}. \]Evaluating a Complex Function
Evaluation works much as it does for real functions.
Suppose
\[ f(z)=z^2+1. \]At
\[ z=2+i, \]we have
\[ f(2+i)=(2+i)^2+1. \]Expand:
\[ (2+i)^2=4+4i+i^2=3+4i. \]Therefore,
\[ \boxed{f(2+i)=4+4i}. \]The Domain of a Complex Function
The domain of a complex function is the set of complex inputs for which the rule is defined.
For example,
\[ f(z)=z^2+3z-1 \]is defined for every complex number.
Thus,
\[ \boxed{\operatorname{Dom}(f)=\mathbb{C}}. \]Domain Restrictions
Complex functions can have restrictions just as real functions do.
Consider
\[ f(z)=\frac{1}{z-2}. \]The denominator cannot equal zero, so
\[ z\neq2. \]Therefore,
\[ \boxed{\operatorname{Dom}(f)=\mathbb{C}\setminus\{2\}}. \]Rational Complex Functions
A rational complex function has the form
\[ f(z)=\frac{P(z)}{Q(z)}, \]where \(P\) and \(Q\) are polynomials.
Its domain excludes the zeros of \(Q\).
For example,
\[ f(z)=\frac{z+1}{z^2+1}. \]Since
\[ z^2+1=(z-i)(z+i), \]the excluded points are
\[ \boxed{z=i,\qquad z=-i}. \]A Complex Function as a Mapping
A complex function may be viewed as a rule that sends each input point
\[ z=x+iy \]to an output point
\[ w=f(z)=u+iv. \]It is common to distinguish the input and output planes:
\[ \boxed{z\text{-plane}\longrightarrow w\text{-plane}}. \]The function transforms geometry from one plane into geometry in the other.
Translation: \(f(z)=z+c\)
Let
\[ c=a+bi. \]Then
\[ f(z)=z+c \]shifts every point by the same complex displacement.
Geometrically, this is a translation by the vector
\[ \boxed{(a,b)}. \]Example: Translation
Consider
\[ f(z)=z+(3-2i). \]The point
\[ z=1+i \]maps to
\[ f(1+i) = 1+i+3-2i. \]Therefore,
\[ \boxed{f(1+i)=4-i}. \]Every point is shifted \(3\) units to the right and \(2\) units downward.
Real Scaling: \(f(z)=az\)
If \(a>0\) is real, then
\[ f(z)=az \]multiplies every distance from the origin by \(a\) without changing direction.
Thus,
\[ \boxed{|f(z)|=a|z|}. \]Complex Scaling and Rotation
Let
\[ c=\rho e^{i\phi}. \]Then
\[ f(z)=cz \]scales every distance by \(\rho\) and rotates every direction by \(\phi\).
If
\[ z=re^{i\theta}, \]then
\[ f(z) = \rho r e^{i(\theta+\phi)}. \]Therefore, multiplication by a fixed nonzero complex number is a combined dilation and rotation about the origin.
Example: Rotation by \(i\)
Consider
\[ f(z)=iz. \]Because
\[ i=e^{i\pi/2}, \]the function rotates every point counterclockwise by
\[ \boxed{\frac{\pi}{2}}. \]For example,
\[ f(2+i)=i(2+i)=-1+2i. \]Affine Complex Functions
A function of the form
\[ \boxed{f(z)=az+b} \]with \(a\neq0\) combines:
- scaling by \(|a|\);
- rotation by \(\arg a\);
- translation by \(b\).
These transformations provide an important geometric model for more advanced complex mappings.
The Conjugation Function
Consider
\[ f(z)=\overline{z}. \]If
\[ z=x+iy, \]then
\[ f(z)=x-iy. \]Geometrically, this reflects every point across the real axis.
Thus,
\[ \boxed{(x,y)\mapsto(x,-y)}. \]The Modulus Function
The function
\[ f(z)=|z| \]takes a complex input but produces a real, nonnegative output.
Thus its natural codomain may be written
\[ f:\mathbb{C}\to[0,\infty). \]If \(z=x+iy\), then
\[ \boxed{f(z)=\sqrt{x^2+y^2}}. \]The Real-Part Function
The function
\[ f(z)=\operatorname{Re}(z) \]sends
\[ x+iy \]to
\[ x. \]Therefore,
\[ \boxed{f(x+iy)=x}. \]It projects the complex plane onto the real axis.
The Imaginary-Part Function
The function
\[ f(z)=\operatorname{Im}(z) \]sends
\[ x+iy \]to
\[ y. \]Thus,
\[ \boxed{f(x+iy)=y}. \]The Reciprocal Function
Consider
\[ f(z)=\frac1z. \]Its domain is
\[ \mathbb{C}\setminus\{0\}. \]If
\[ z=re^{i\theta}, \]then
\[ f(z)=\frac1r e^{-i\theta}. \]Thus the reciprocal:
- replaces the modulus \(r\) by \(1/r\);
- reverses the sign of the argument.
Reciprocal Function in Rectangular Coordinates
If
\[ z=x+iy\neq0, \]then
\[ \frac1z = \frac{x-iy}{x^2+y^2}. \]Therefore,
\[ \boxed{ u(x,y)=\frac{x}{x^2+y^2} } \]and
\[ \boxed{ v(x,y)=\frac{-y}{x^2+y^2} }. \]The Squaring Function Geometrically
Consider
\[ f(z)=z^2. \]If
\[ z=re^{i\theta}, \]then
\[ f(z)=r^2e^{i2\theta}. \]Thus squaring:
- squares the modulus;
- doubles the argument.
This makes the geometry of \(z^2\) far easier to understand in polar form.
Example: Squaring a Point
Let
\[ z=2e^{i\pi/6}. \]Then
\[ z^2 = 4e^{i\pi/3}. \]The distance from the origin changes from \(2\) to \(4\), and the angle changes from \(\pi/6\) to \(\pi/3\).
Power Functions
More generally,
\[ f(z)=z^n \]gives
\[ f(re^{i\theta}) = r^ne^{in\theta}. \]Therefore:
\[ \boxed{|f(z)|=|z|^n} \]and
\[ \boxed{\arg f(z)=n\arg z\pmod{2\pi}}. \]Power functions provide some of the clearest examples of complex functions reshaping angles and radial distances.
Complex Polynomial Functions
A polynomial complex function has the form
\[ f(z) = a_nz^n+a_{n-1}z^{n-1}+\cdots+a_1z+a_0, \]where the coefficients \(a_k\) may themselves be complex.
Polynomial functions are defined for every
\[ z\in\mathbb{C}. \]They will provide important examples of analytic functions later in this section.
The Complex Exponential Function
The complex exponential is defined by
\[ f(z)=e^z. \]If
\[ z=x+iy, \]then
\[ e^z = e^{x+iy} = e^x e^{iy}. \]Using Euler's formula,
\[ \boxed{ e^{x+iy} = e^x(\cos y+i\sin y) }. \]Components of the Complex Exponential
From
\[ e^{x+iy} = e^x\cos y + i e^x\sin y, \]we identify
\[ \boxed{u(x,y)=e^x\cos y} \]and
\[ \boxed{v(x,y)=e^x\sin y}. \]The real coordinate \(x\) controls radial scaling, while the imaginary coordinate \(y\) controls angular position.
The Complex Exponential Is Periodic
Because
\[ e^{i(y+2\pi)} = e^{iy}, \]we have
\[ \boxed{ e^{z+2\pi i}=e^z }. \]Thus the complex exponential has period
\[ \boxed{2\pi i}. \]This is a major difference from the real exponential function.
The Complex Exponential Is Not One-to-One
Since
\[ e^{z+2\pi i}=e^z, \]infinitely many complex inputs can produce the same output.
This nonuniqueness is the reason the inverse complex exponential—the complex logarithm—requires branch choices.
A Preview of the Complex Logarithm
If
\[ z=re^{i\theta}\neq0, \]then solutions of
\[ e^w=z \]have the form
\[ w=\ln r+i(\theta+2\pi k). \]Therefore, the complex logarithm is naturally multivalued:
\[ \boxed{ \log z = \ln|z|+i(\arg z) }. \]A single-valued logarithm requires choosing a branch.
Complex Sine and Cosine
Euler's formula allows sine and cosine to be extended naturally to complex inputs.
The definitions are
\[ \boxed{ \cos z = \frac{e^{iz}+e^{-iz}}{2} } \]and
\[ \boxed{ \sin z = \frac{e^{iz}-e^{-iz}}{2i} }. \]These formulas agree with ordinary sine and cosine when \(z\) is real.
Composition of Complex Functions
Complex functions may be composed just like real functions.
If
\[ f(z)=z^2 \]and
\[ g(z)=z+1, \]then
\[ (f\circ g)(z) = f(g(z)) = (z+1)^2. \]Meanwhile,
\[ (g\circ f)(z) = z^2+1. \]As with real functions, composition is generally not commutative.
Inverse Complex Functions
A function can have a true inverse only when its mapping is one-to-one on the chosen domain.
For example,
\[ f(z)=z^2 \]is not one-to-one on all of \(\mathbb{C}\), because
\[ z^2=(-z)^2. \]To define a single-valued square-root inverse, the domain or range must be restricted appropriately.
Such restrictions will later lead to the idea of branches and branch cuts.
Level Sets of the Real and Imaginary Parts
If
\[ f(z)=u(x,y)+iv(x,y), \]then equations such as
\[ u(x,y)=c \]and
\[ v(x,y)=d \]describe curves in the input plane.
These level curves often reveal important geometry in complex mappings.
The Image of a Set
If \(S\) is a set of complex numbers, then its image under \(f\) is
\[ \boxed{ f(S)=\{f(z):z\in S\}. } \]Complex analysis often asks not merely what happens to one point, but what happens to an entire line, circle, disk, sector, or region.
The Preimage of a Set
If \(T\) is a set in the output plane, the preimage is
\[ \boxed{ f^{-1}(T) = \{z:f(z)\in T\}. } \]This notation describes the set of all inputs that map into \(T\).
It does not require \(f\) to possess a single-valued inverse function.
Example: Mapping a Circle by \(f(z)=2z\)
Consider the circle
\[ |z|=3. \]Under
\[ w=2z, \]the output modulus is
\[ |w|=|2z|=2|z|. \]Therefore,
\[ |w|=6. \]So the circle of radius \(3\) maps to the circle
\[ \boxed{|w|=6}. \]Example: Mapping a Circle by Translation
Let
\[ w=z+(2-i). \]Suppose the input set is
\[ |z|=4. \]Since
\[ z=w-(2-i), \]the image satisfies
\[ |w-(2-i)|=4. \]Thus the circle remains a circle with the same radius but moves to the center
\[ \boxed{2-i}. \]Example: Mapping the Unit Circle by \(f(z)=z^2\)
On the unit circle,
\[ z=e^{i\theta}. \]Then
\[ w=z^2=e^{i2\theta}. \]Because
\[ |w|=1, \]the unit circle maps onto itself.
However, as \(\theta\) travels once around the input circle, \(2\theta\) travels twice around the output circle.
Different Inputs Can Produce the Same Output
For
\[ f(z)=z^2, \]the inputs
\[ z \]and
\[ -z \]produce the same output:
\[ f(z)=f(-z). \]This illustrates how a complex mapping may fold or overlap parts of the plane.
Fixed Points of a Complex Function
A fixed point is a point satisfying
\[ \boxed{f(z)=z}. \]For example, if
\[ f(z)=z^2, \]then fixed points satisfy
\[ z^2=z. \]Factor:
\[ z(z-1)=0. \]Therefore,
\[ \boxed{z=0,\qquad z=1}. \]Zeros of a Complex Function
A zero of \(f\) is a complex number \(z_0\) satisfying
\[ \boxed{f(z_0)=0}. \]For a polynomial, zeros are exactly its complex roots.
For example,
\[ f(z)=z^2+1 \]has zeros
\[ \boxed{z=i,\qquad z=-i}. \]A First Look at Singular Points
Some complex functions fail to be defined or fail to behave regularly at particular points.
For example,
\[ f(z)=\frac1z \]is undefined at
\[ z=0. \]Such exceptional points will later be studied systematically under singularities.
Complex Functions Differ from Real Functions
A real function of one real variable receives input from a line.
A complex function receives input from a plane.
This means that when we later study limits and derivatives, an input can approach a point from infinitely many directions.
This geometric freedom makes complex differentiability a much stronger condition than ordinary real differentiability.
A Complex Function Contains Two Real Functions
Writing
\[ f(z)=u(x,y)+iv(x,y) \]makes clear that a complex function may be viewed as the real vector-valued mapping
\[ \boxed{ (x,y)\mapsto(u(x,y),v(x,y)). } \]Thus tools from multivariable calculus can help us analyze complex functions.
But the complex-number structure imposes additional relationships that will become important in complex differentiation.
Example: Component Functions of \(\frac1z\)
Let
\[ f(z)=\frac1z, \qquad z=x+iy\neq0. \]Multiply numerator and denominator by \(x-iy\):
\[ \frac1{x+iy} = \frac{x-iy}{x^2+y^2}. \]Therefore,
\[ f(z) = \frac{x}{x^2+y^2} - i\frac{y}{x^2+y^2}. \]Hence,
\[ \boxed{ u(x,y)=\frac{x}{x^2+y^2} } \]and
\[ \boxed{ v(x,y)=\frac{-y}{x^2+y^2}. } \]Example: Component Functions of \(e^z\)
Let
\[ z=x+iy. \]Then
\[ e^z = e^x(\cos y+i\sin y). \]Therefore,
\[ \boxed{ u(x,y)=e^x\cos y } \]and
\[ \boxed{ v(x,y)=e^x\sin y. } \]Useful Basic Complex Functions
- \(f(z)=c\): constant function;
- \(f(z)=z\): identity function;
- \(f(z)=az+b\): affine function;
- \(f(z)=z^n\): power function;
- \(f(z)=1/z\): reciprocal function;
- \(f(z)=\overline{z}\): conjugation;
- \(f(z)=|z|\): modulus function;
- \(f(z)=e^z\): complex exponential;
- \(f(z)=\sin z\): complex sine;
- \(f(z)=\cos z\): complex cosine.
Choosing a Useful Representation
Different forms of a complex number reveal different aspects of a function.
Rectangular form
\[ z=x+iy \]is useful when separating real and imaginary parts.
Polar or exponential form
\[ z=re^{i\theta} \]is especially useful when analyzing powers, roots, scaling, rotation, and angular behavior.
Common Error: Treating \(z\) and \(\overline{z}\) as Independent
If
\[ z=x+iy, \]then
\[ \overline{z}=x-iy. \]The two quantities are linked by the same real coordinates \(x\) and \(y\).
A function containing \(\overline{z}\) therefore has different structural properties from one depending only on powers of \(z\).
Common Error: Assuming Every Complex Function Has Complex Outputs
A function may accept complex inputs but return real outputs.
Examples include
\[ |z|, \qquad \operatorname{Re}(z), \qquad \operatorname{Im}(z). \]Always distinguish the domain from the codomain.
Common Error: Ignoring Domain Restrictions
For
\[ f(z)=\frac1{z-a}, \]the point
\[ z=a \]must be excluded.
Domain restrictions matter later when studying limits, continuity, poles, and contour integrals.
Common Error: Thinking of a Complex Function as an Ordinary Graph
A real function \(y=f(x)\) can often be drawn directly in a two-dimensional plane.
A general complex function has two real input coordinates and two real output coordinates.
A literal graph would therefore require four real dimensions.
Instead, complex functions are often visualized through mappings, transformed grids, level curves, color plots, or separate input and output planes.
A Reliable Procedure for Finding Component Functions
- Write \(z=x+iy\).
- Substitute \(x+iy\) into the function.
- Expand and simplify using \(i^2=-1\).
- Collect all real terms.
- Collect all coefficients of \(i\).
- Write the result as \(u(x,y)+iv(x,y)\).
- Identify \(u\) and \(v\).
A Reliable Procedure for Studying a Mapping
- Identify the domain and any excluded points.
- Choose rectangular or polar form according to the function.
- Track what happens to modulus.
- Track what happens to argument.
- Test representative points.
- Analyze simple sets such as lines, circles, or rays.
- Check whether multiple inputs can map to the same output.
Worked Example: An Affine Mapping
Consider
\[ f(z)=(1+i)z+2. \]Since
\[ 1+i=\sqrt2\,e^{i\pi/4}, \]multiplication by \(1+i\):
- scales distances by \(\sqrt2\);
- rotates points by \(\pi/4\).
The final \(+2\) translates the result two units to the right.
Thus the full mapping is a scaling, rotation, and translation.
Worked Example: Analyze \(f(z)=z^2\)
Write
\[ z=re^{i\theta}. \]Then
\[ f(z)=r^2e^{i2\theta}. \]Therefore:
- the modulus \(r\) becomes \(r^2\);
- the angle \(\theta\) becomes \(2\theta\);
- the unit circle maps onto itself;
- each nonzero output generally has two square-root preimages.
The function therefore stretches radially and doubles angles.
Worked Example: Analyze \(f(z)=1/z\)
Write
\[ z=re^{i\theta}. \]Then
\[ \frac1z = \frac1r e^{-i\theta}. \]Therefore:
- points outside the unit circle move inside it;
- points inside the unit circle move outside it;
- points on the unit circle remain on the unit circle;
- angles reverse sign.
Worked Example: Find \(u\) and \(v\)
Let
\[ f(z)=z^2+2z. \]Write
\[ z=x+iy. \]Then
\[ f(z) = (x+iy)^2+2(x+iy). \]Expand:
\[ x^2-y^2+2ixy+2x+2iy. \]Group real and imaginary terms:
\[ f(z) = (x^2-y^2+2x) + i(2xy+2y). \]Therefore,
\[ \boxed{ u(x,y)=x^2-y^2+2x } \]and
\[ \boxed{ v(x,y)=2xy+2y. } \]Practice 1: Evaluate a Function
If
\[ f(z)=z^2-2z+3, \]find
\[ f(1+i). \]Practice 2: Find the Domain
Find the domain of
\[ f(z)=\frac{z+1}{z^2-4}. \]Practice 3: Find Component Functions
Write
\[ f(z)=z^2-1 \]in the form
\[ u(x,y)+iv(x,y). \]Practice 4: Translation
Under
\[ f(z)=z-3+2i, \]where does
\[ z=4-i \]map?
Practice 5: Rotation and Scaling
Describe geometrically the function
\[ f(z)=2iz. \]Practice 6: Power Mapping
If
\[ z=3e^{i\pi/5}, \]find \(f(z)\) for
\[ f(z)=z^3. \]Practice 7: Reciprocal Mapping
If
\[ z=4e^{i\pi/3}, \]find
\[ \frac1z. \]Practice 8: Complex Exponential
Write
\[ e^{2+i\pi/2} \]in rectangular form.
Practice 9: Fixed Points
Find the fixed points of
\[ f(z)=z^2-2. \]Practice 10: Image of a Circle
Under
\[ w=3z, \]determine the image of
\[ |z|=2. \]Practice Answers
Practice 1
Since
\[ (1+i)^2=2i, \]we have
\[ f(1+i) = 2i-2(1+i)+3. \] \[ =2i-2-2i+3. \]Therefore,
\[ \boxed{f(1+i)=1}. \]Practice 2
The denominator is zero when
\[ z^2-4=0, \]so
\[ z=\pm2. \]Therefore,
\[ \boxed{ \operatorname{Dom}(f) = \mathbb{C}\setminus\{-2,2\}. } \]Practice 3
\[ f(z) = (x+iy)^2-1 \] \[ = x^2-y^2-1+i(2xy). \]Therefore,
\[ \boxed{ u(x,y)=x^2-y^2-1 } \]and
\[ \boxed{ v(x,y)=2xy. } \]Practice 4
\[ f(4-i) = 4-i-3+2i. \]Therefore,
\[ \boxed{1+i}. \]Practice 5
Multiplication by \(2i\) scales distances by
\[ |2i|=2 \]and rotates by
\[ \frac{\pi}{2}. \]Thus the mapping is a dilation by factor \(2\) followed by a \(90^\circ\) counterclockwise rotation.
Practice 6
\[ f(z) = \left(3e^{i\pi/5}\right)^3. \] \[ = 27e^{i3\pi/5}. \]Therefore,
\[ \boxed{27e^{i3\pi/5}}. \]Practice 7
\[ \frac1z = \frac14e^{-i\pi/3}. \]Therefore,
\[ \boxed{ \frac14e^{-i\pi/3} }. \]Practice 8
\[ e^{2+i\pi/2} = e^2e^{i\pi/2}. \]Since
\[ e^{i\pi/2}=i, \]we obtain
\[ \boxed{ie^2}. \]Practice 9
Fixed points satisfy
\[ z^2-2=z. \]Rearranging,
\[ z^2-z-2=0. \]Factor:
\[ (z-2)(z+1)=0. \]Therefore,
\[ \boxed{z=2,\qquad z=-1}. \]Practice 10
Since
\[ |w|=|3z|=3|z|, \]and
\[ |z|=2, \]we obtain
\[ \boxed{|w|=6}. \]Checking Strategies
- Substitute a simple point such as \(z=0\), \(1\), or \(i\) to check an algebraic formula.
- When finding \(u\) and \(v\), verify that every remaining coefficient is real.
- When using polar form, check modulus and argument separately.
- For a claimed image of a circle or line, test several representative points.
- Always verify domain exclusions before performing later analysis.
Connections to Earlier Mathematics
Complex functions connect several major mathematical ideas:
- Functions: inputs, outputs, composition, inverses, zeros, and fixed points.
- Coordinate geometry: transformations of points and regions.
- Linear algebra: scaling, rotation, reflection, and two-dimensional mappings.
- Multivariable calculus: the component functions \(u(x,y)\) and \(v(x,y)\).
- Trigonometry and exponentials: polar and exponential descriptions of geometric transformations.
What to Remember
A complex function may be written
\[ \boxed{f:\mathbb{C}\to\mathbb{C}}. \]If
\[ z=x+iy, \]then
\[ \boxed{ f(z)=u(x,y)+iv(x,y). } \]Complex functions can be studied both algebraically and geometrically.
Important examples include:
\[ z^n,\qquad \frac1z,\qquad e^z,\qquad \overline{z},\qquad az+b. \]The central viewpoint is: a complex function transforms the complex plane.
Where We Go Next
To analyze a complex function rigorously, we must understand what happens when an input \(z\) approaches a point in the complex plane.
Unlike the real line, the complex plane allows infinitely many directions of approach.
The next lesson, Complex Limits and Continuity, develops the meaning of \[ \lim_{z\to z_0}f(z), \] explains why every path of approach must agree, and establishes the continuity framework required for complex differentiation.