The Central Idea
Let
\[ f:\mathbb{C}\to\mathbb{C}. \]The derivative of \(f\) at \(z_0\) is defined by
\[ \boxed{ f'(z_0) = \lim_{h\to0} \frac{f(z_0+h)-f(z_0)}{h} }. \]This formula looks almost identical to the real derivative.
The major difference is that
\[ h\in\mathbb{C}, \]so \(h\to0\) through every possible direction in the plane.
Why Complex Differentiability Is Stronger
In real calculus, the increment \(h\) approaches \(0\) along a line.
In complex analysis, \(h\) may approach \(0\) through:
- the real axis;
- the imaginary axis;
- any other line through the origin;
- curved paths;
- spirals or other trajectories.
The difference quotient must approach the same complex number along all of them.
Differentiability at a Point
A function is complex differentiable at \(z_0\) if the limit
\[ \lim_{h\to0} \frac{f(z_0+h)-f(z_0)}{h} \]exists as a finite complex number.
When the limit exists, its value is denoted
\[ \boxed{f'(z_0)}. \]Differentiability on a Region
A function is complex differentiable on a set if it is differentiable at every point of that set where the derivative is being considered.
Later, when differentiability holds throughout an open region, we will use the stronger language of analytic or holomorphic functions.
Example: Differentiate \(f(z)=z^2\)
Start with
\[ f'(z) = \lim_{h\to0} \frac{(z+h)^2-z^2}{h}. \]Expand:
\[ (z+h)^2-z^2 = 2zh+h^2. \]Therefore,
\[ f'(z) = \lim_{h\to0} \frac{2zh+h^2}{h}. \]For \(h\neq0\),
\[ \frac{2zh+h^2}{h} = 2z+h. \]Hence,
\[ \boxed{ f'(z)=2z. } \]Derivative of a Constant
If
\[ f(z)=c \]for a constant \(c\in\mathbb{C}\), then
\[ f'(z) = \lim_{h\to0} \frac{c-c}{h}. \]Therefore,
\[ \boxed{f'(z)=0}. \]Derivative of the Identity Function
For
\[ f(z)=z, \]the difference quotient is
\[ \frac{(z+h)-z}{h}=1. \]Thus,
\[ \boxed{f'(z)=1}. \]The Power Rule
For a positive integer \(n\),
\[ f(z)=z^n \]satisfies
\[ \boxed{ \frac{d}{dz}z^n=nz^{n-1}. } \]The familiar real-variable power rule carries over directly to complex polynomials.
Polynomial Derivatives
If
\[ P(z) = a_nz^n+a_{n-1}z^{n-1}+\cdots+a_1z+a_0, \]then
\[ \boxed{ P'(z) = na_nz^{n-1} +(n-1)a_{n-1}z^{n-2} +\cdots+a_1. } \]Complex polynomial differentiation therefore looks exactly like polynomial differentiation over the real numbers.
Example: Differentiate a Complex Polynomial
Let
\[ f(z)=3z^4-(2+i)z^2+5z-7i. \]Differentiate term by term:
\[ f'(z) = 12z^3-2(2+i)z+5. \]Therefore,
\[ \boxed{ f'(z)=12z^3-(4+2i)z+5. } \]Linearity of Differentiation
If \(f\) and \(g\) are complex differentiable at \(z\), and \(a,b\in\mathbb{C}\), then
\[ \boxed{ (af+bg)'(z) = af'(z)+bg'(z). } \]Thus sums and constant multiples differentiate exactly as expected.
The Product Rule
If \(f\) and \(g\) are complex differentiable, then
\[ \boxed{ (fg)'(z) = f'(z)g(z)+f(z)g'(z). } \]The algebraic proof is the same as in real calculus because it depends only on limit laws and distributivity.
The Quotient Rule
If \(f\) and \(g\) are complex differentiable and
\[ g(z)\neq0, \]then
\[ \boxed{ \left(\frac{f}{g}\right)'(z) = \frac{f'(z)g(z)-f(z)g'(z)}{[g(z)]^2}. } \]The Chain Rule
If \(g\) is complex differentiable at \(z\) and \(f\) is complex differentiable at \(g(z)\), then
\[ \boxed{ (f\circ g)'(z) = f'(g(z))g'(z). } \]Composition therefore preserves the familiar derivative structure of real calculus.
Derivative of the Reciprocal Function
For
\[ f(z)=\frac1z, \qquad z\neq0, \]the power or quotient rule gives
\[ \boxed{ f'(z)=-\frac1{z^2}. } \]The function is complex differentiable everywhere except at its excluded point \(z=0\).
Rational Functions
A rational function
\[ f(z)=\frac{P(z)}{Q(z)} \]is complex differentiable wherever
\[ Q(z)\neq0. \]Thus rational functions behave smoothly throughout their domains, with exceptional behavior occurring only at zeros of the denominator.
Differentiability Implies Continuity
If \(f\) is complex differentiable at \(z_0\), then \(f\) is continuous at \(z_0\).
To see why, write
\[ f(z)-f(z_0) = \frac{f(z)-f(z_0)}{z-z_0}(z-z_0) \]for \(z\neq z_0\).
As \(z\to z_0\),
\[ \frac{f(z)-f(z_0)}{z-z_0} \to f'(z_0) \]while
\[ z-z_0\to0. \]Therefore,
\[ f(z)-f(z_0)\to0. \]Hence,
\[ \boxed{ f(z)\to f(z_0). } \]Continuity Does Not Imply Complex Differentiability
The converse is false.
A function can be continuous everywhere and still fail to possess a complex derivative.
Two important examples are
\[ f(z)=\overline{z} \]and
\[ f(z)=|z|. \]Both are continuous, but neither is complex differentiable anywhere.
Example: Why \(f(z)=\overline{z}\) Is Not Complex Differentiable
Consider the difference quotient at \(z_0\):
\[ \frac{\overline{z_0+h}-\overline{z_0}}{h} = \frac{\overline{h}}{h}. \]Approach \(h=0\) along the real axis:
\[ h=t, \]so
\[ \frac{\overline{h}}{h}=1. \]Approach along the imaginary axis:
\[ h=it, \]so
\[ \frac{\overline{h}}{h} = \frac{-it}{it} = -1. \]The two directional limits disagree.
Therefore,
\[ \boxed{ f(z)=\overline{z} \text{ is nowhere complex differentiable}. } \]Example: Why \(f(z)=|z|\) Is Not Complex Differentiable
At \(z=0\), the difference quotient is
\[ \frac{|h|}{h}. \]Along positive real \(h=t>0\),
\[ \frac{|h|}{h}=1. \]Along positive imaginary \(h=it\),
\[ \frac{|h|}{h} = \frac{t}{it} = -i. \]The limits disagree, so \(f\) is not complex differentiable at \(0\).
A similar directional argument shows that \(|z|\) is not complex differentiable at any nonzero point either.
Example: The Real-Part Function
Let
\[ f(z)=\operatorname{Re}(z). \]At any point \(z_0\), the difference quotient becomes
\[ \frac{\operatorname{Re}(h)}{h}. \]Along \(h=t\),
\[ \frac{\operatorname{Re}(t)}{t}=1. \]Along \(h=it\),
\[ \frac{\operatorname{Re}(it)}{it}=0. \]Therefore,
\[ \boxed{ \operatorname{Re}(z) \text{ is nowhere complex differentiable}. } \]Example: The Imaginary-Part Function
For
\[ f(z)=\operatorname{Im}(z), \]the difference quotient also depends on the direction of \(h\).
Along the real axis, the quotient is \(0\), while along the imaginary axis it gives a different value.
Therefore,
\[ \boxed{ \operatorname{Im}(z) \text{ is nowhere complex differentiable}. } \]Writing a Complex Function in Components
Let
\[ z=x+iy \]and write
\[ f(z)=u(x,y)+iv(x,y). \]The complex derivative places conditions on how the two real component functions \(u\) and \(v\) change in the \(x\)- and \(y\)-directions.
This is where multivariable calculus and complex analysis meet.
Approaching Through Real Increments
Suppose \(h\) is real:
\[ h=\Delta x. \]Then
\[ \frac{f(z+h)-f(z)}{h} \]measures change in the \(x\)-direction.
When the relevant partial derivatives exist, the limiting value becomes
\[ \boxed{ u_x+iv_x. } \]Approaching Through Imaginary Increments
Now let
\[ h=i\Delta y. \]Then the same complex difference quotient measures change in the \(y\)-direction.
Under appropriate differentiability assumptions, the limiting value becomes
\[ \boxed{ v_y-iu_y. } \]Directional Consistency
If the complex derivative exists, the real-direction and imaginary-direction limits must agree.
Therefore,
\[ u_x+iv_x = v_y-iu_y. \]Matching real and imaginary parts gives
\[ u_x=v_y \]and
\[ v_x=-u_y. \]These are the relationships developed systematically in the next lesson.
A Preview of the Cauchy–Riemann Relationships
The equations
\[ \boxed{u_x=v_y} \]and
\[ \boxed{u_y=-v_x} \]are called the Cauchy–Riemann relationships.
They are necessary for complex differentiability under the usual partial derivative assumptions.
With appropriate continuity of the partial derivatives, they also provide a practical sufficient test for differentiability.
Recovering the Derivative from Component Functions
When \(f=u+iv\) is complex differentiable,
\[ \boxed{ f'(z)=u_x+iv_x. } \]Using the Cauchy–Riemann relationships, the same derivative may also be written
\[ \boxed{ f'(z)=v_y-iu_y. } \]These equivalent forms are useful for checking calculations.
Example: Components of \(f(z)=z^2\)
Since
\[ z=x+iy, \]we have
\[ z^2=x^2-y^2+i(2xy). \]Thus,
\[ u(x,y)=x^2-y^2 \]and
\[ v(x,y)=2xy. \]Then
\[ u_x=2x, \qquad v_x=2y. \]Therefore,
\[ f'(z) = 2x+i(2y) = 2(x+iy). \]Hence,
\[ \boxed{f'(z)=2z}. \]The Derivative as Local Complex Multiplication
Near a point \(z_0\), a differentiable complex function behaves approximately like
\[ f(z_0+h) \approx f(z_0)+f'(z_0)h. \]Equivalently,
\[ \boxed{ f(z)-f(z_0) \approx f'(z_0)(z-z_0). } \]Thus the derivative acts locally as multiplication by the complex number \(f'(z_0)\).
Local Scaling and Rotation
Write
\[ f'(z_0) = \rho e^{i\phi}. \]Then, for very small increments \(h\),
\[ f(z_0+h)-f(z_0) \approx \rho e^{i\phi}h. \]Therefore, when
\[ f'(z_0)\neq0, \]the function locally behaves like:
- a scaling by \(\rho=|f'(z_0)|\);
- a rotation by \(\phi=\arg f'(z_0)\).
A Preview of Angle Preservation
Multiplication by a nonzero complex number rotates every direction by the same angle and scales every length by the same factor.
Therefore, when
\[ f'(z_0)\neq0, \]a complex differentiable function often preserves local angles between intersecting curves.
This is the foundation of conformal mapping, studied later in this section.
What If \(f'(z_0)=0\)?
If
\[ f'(z_0)=0, \]the first-order linear approximation collapses.
Higher-order terms then control the local behavior.
For example,
\[ f(z)=z^2 \]has
\[ f'(0)=0. \]Near the origin, the squaring operation doubles angles rather than behaving like a nonzero rotation-and-scaling map.
Derivative of the Complex Exponential
The complex exponential satisfies the same derivative rule as the real exponential:
\[ \boxed{ \frac{d}{dz}e^z=e^z. } \]This compatibility is one reason the exponential function plays such a central role in complex analysis.
Checking \(e^z\) Through Its Components
Recall
\[ e^{x+iy} = e^x\cos y + i e^x\sin y. \]Thus
\[ u=e^x\cos y \]and
\[ v=e^x\sin y. \]Differentiate with respect to \(x\):
\[ u_x=e^x\cos y, \qquad v_x=e^x\sin y. \]Therefore,
\[ f'(z) = u_x+iv_x = e^x(\cos y+i\sin y). \]Hence,
\[ \boxed{f'(z)=e^z}. \]Complex Sine and Cosine Derivatives
The familiar trigonometric derivative rules remain valid:
\[ \boxed{ \frac{d}{dz}\sin z=\cos z } \]and
\[ \boxed{ \frac{d}{dz}\cos z=-\sin z. } \]These formulas can be derived efficiently from the exponential definitions of sine and cosine.
Higher Complex Derivatives
If \(f'\) is itself complex differentiable, we define
\[ f''(z) = \frac{d}{dz}f'(z). \]Repeating this process gives
\[ f^{(3)}(z), \quad f^{(4)}(z), \quad \ldots \]Analytic functions will turn out to possess derivatives of every order.
Example: Higher Derivatives
Let
\[ f(z)=z^4. \]Then
\[ f'(z)=4z^3, \] \[ f''(z)=12z^2, \] \[ f^{(3)}(z)=24z, \] \[ f^{(4)}(z)=24, \]and
\[ f^{(5)}(z)=0. \]Example: Chain Rule
Differentiate
\[ f(z)=e^{z^2}. \]Let
\[ g(z)=z^2. \]Then
\[ f(z)=e^{g(z)}. \]The chain rule gives
\[ f'(z) = e^{z^2}(2z). \]Therefore,
\[ \boxed{ f'(z)=2ze^{z^2}. } \]Example: Product Rule
Differentiate
\[ f(z)=z^2e^z. \]Using the product rule,
\[ f'(z) = (2z)e^z+z^2e^z. \]Factor:
\[ \boxed{ f'(z)=e^z(z^2+2z). } \]Example: Quotient Rule
Differentiate
\[ f(z)=\frac{z+1}{z-1}. \]For \(z\neq1\),
\[ f'(z) = \frac{(1)(z-1)-(z+1)(1)}{(z-1)^2}. \]Simplify:
\[ f'(z) = \frac{z-1-z-1}{(z-1)^2}. \]Therefore,
\[ \boxed{ f'(z) = -\frac{2}{(z-1)^2}. } \]Differentiable at One Point Versus in a Neighborhood
A function may be complex differentiable at a single isolated point without being differentiable throughout any surrounding disk.
This distinction matters.
Complex analysis reserves especially strong conclusions for functions that are differentiable throughout open regions, not merely at isolated points.
Example: A Function Differentiable Only at One Point
Consider
\[ f(z)=z\overline{z}=|z|^2. \]At \(z=0\),
\[ \frac{f(h)-f(0)}{h} = \frac{|h|^2}{h}. \]Since
\[ |h|^2=h\overline{h}, \]for \(h\neq0\),
\[ \frac{|h|^2}{h} = \overline{h}. \]As \(h\to0\),
\[ \overline{h}\to0. \]Therefore,
\[ \boxed{f'(0)=0}. \]Away from \(0\), however, the function fails the complex differentiability conditions.
An Important Nuance
The example
\[ f(z)=z\overline{z} \]shows why one must distinguish:
- differentiability at a single point;
- differentiability on an open region.
The second condition is far more powerful and leads to the theory of analytic functions.
An Equivalent Difference Quotient
Instead of using \(h\), we may write
\[ \boxed{ f'(z_0) = \lim_{z\to z_0} \frac{f(z)-f(z_0)}{z-z_0}. } \]This form makes the connection with complex limits especially clear.
What Replaces the Ordinary Tangent Slope?
In real calculus, \(f'(x_0)\) is often interpreted as the slope of a tangent line.
A general complex function is not naturally drawn as a two-dimensional graph, so a single real slope is no longer the primary interpretation.
Instead, \(f'(z_0)\) is best viewed as a local linear transformation:
\[ \boxed{ h\mapsto f'(z_0)h. } \]This complex multiplication records both local scale and local rotation.
The Magnitude of the Derivative
If
\[ f'(z_0)\neq0, \]then
\[ |f'(z_0)| \]gives the local first-order scale factor.
Small lengths near \(z_0\) are approximately multiplied by
\[ \boxed{|f'(z_0)|}. \]The Argument of the Derivative
If
\[ f'(z_0)\neq0, \]then
\[ \arg f'(z_0) \]gives the local first-order rotation angle.
Thus the derivative contains geometric information unavailable from an ordinary real slope alone.
Common Error: Checking Only Real Increments
If the difference quotient has a limit when
\[ h\in\mathbb{R}, \]that does not prove the complex derivative exists.
The same quotient must agree for imaginary and all other complex directions.
Common Error: Assuming Partial Derivatives Alone Are Enough
The existence of
\[ u_x,\quad u_y,\quad v_x,\quad v_y \]does not by itself guarantee complex differentiability.
The component changes must satisfy special compatibility conditions, and suitable regularity assumptions are needed for common sufficient tests.
Common Error: Assuming Continuity Guarantees a Derivative
Continuity is necessary for differentiability, but it is not sufficient.
For example,
\[ f(z)=\overline{z} \]is continuous everywhere but complex differentiable nowhere.
Common Error: Applying Real Derivative Rules to Nonanalytic Expressions
Rules such as the power, product, quotient, and chain rules apply when the functions involved are already complex differentiable.
An expression involving
\[ \overline{z}, \quad |z|, \quad \operatorname{Re}(z), \quad \operatorname{Im}(z) \]should not automatically be differentiated as if \(z\) were an ordinary real variable.
Common Error: Canceling \(h\) Before Establishing \(h\neq0\)
In the difference quotient,
\[ \frac{f(z+h)-f(z)}{h}, \]cancellation is performed only for
\[ h\neq0. \]The derivative is obtained by taking the limit as \(h\to0\), not by setting \(h=0\) inside the quotient.
A Reliable Procedure: Differentiate from the Definition
- Write the difference quotient.
- Expand or simplify the numerator.
- Factor out \(h\) when possible.
- Cancel only while \(h\neq0\).
- Take the complex limit as \(h\to0\).
- Check whether the result is independent of the direction of \(h\).
- If different directions disagree, conclude that the derivative does not exist.
A Reliable Procedure: Use Component Functions
- Write \(z=x+iy\).
- Rewrite \(f(z)\) as \(u(x,y)+iv(x,y)\).
- Compute \(u_x,u_y,v_x,v_y\).
- Check the compatibility relationships between those partial derivatives.
- Verify any regularity assumptions required by the differentiability test being used.
- When differentiability holds, compute \(f'(z)=u_x+iv_x\).
- Check with \(f'(z)=v_y-iu_y\) when useful.
A Reliable Procedure: Use Derivative Rules
- Confirm that the basic functions involved are complex differentiable on the region.
- Identify sums, products, quotients, or compositions.
- Apply the corresponding derivative rules.
- Respect domain restrictions such as zeros of denominators.
- Simplify the result.
- Check special points where the formula may fail.
Worked Example: Differentiate \(f(z)=z^3\) from the Definition
Start with
\[ f'(z) = \lim_{h\to0} \frac{(z+h)^3-z^3}{h}. \]Expand:
\[ (z+h)^3 = z^3+3z^2h+3zh^2+h^3. \]Therefore,
\[ f'(z) = \lim_{h\to0} \frac{3z^2h+3zh^2+h^3}{h}. \]For \(h\neq0\),
\[ = \lim_{h\to0} \left( 3z^2+3zh+h^2 \right). \]Hence,
\[ \boxed{ f'(z)=3z^2. } \]Worked Example: A Direction-Dependent Quotient
Consider
\[ f(z)=\overline{z} \]at \(z=0\).
The quotient is
\[ \frac{\overline{h}}{h}. \]Along
\[ h=t, \]it equals \(1\).
Along
\[ h=it, \]it equals \(-1\).
Because the limit depends on direction,
\[ \boxed{ f'(0)\text{ does not exist}. } \]Worked Example: Differentiate Through Components
Let
\[ f(z)=z^2+2z. \]With \(z=x+iy\),
\[ f(z) = x^2-y^2+2x + i(2xy+2y). \]Therefore,
\[ u=x^2-y^2+2x \]and
\[ v=2xy+2y. \]Differentiate with respect to \(x\):
\[ u_x=2x+2, \qquad v_x=2y. \]Thus
\[ f'(z) = 2x+2+i(2y). \]Since \(z=x+iy\),
\[ \boxed{ f'(z)=2z+2. } \]Worked Example: Differentiability of \(z\overline{z}\) at the Origin
Let
\[ f(z)=z\overline{z}. \]At the origin,
\[ f'(0) = \lim_{h\to0} \frac{h\overline{h}}{h}. \]For \(h\neq0\),
\[ \frac{h\overline{h}}{h} = \overline{h}. \]Therefore,
\[ \boxed{ f'(0)=0. } \]This function is differentiable at \(0\), but not on any open neighborhood of \(0\).
Practice 1: Definition
Use the definition to differentiate
\[ f(z)=z^2+3z. \]Practice 2: Polynomial Rule
Differentiate
\[ f(z)=5z^4-2z^3+7z-i. \]Practice 3: Product Rule
Differentiate
\[ f(z)=z^3e^z. \]Practice 4: Quotient Rule
Differentiate
\[ f(z)=\frac{z^2+1}{z}, \qquad z\neq0. \]Practice 5: Chain Rule
Differentiate
\[ f(z)=e^{3z}. \]Practice 6: Test Differentiability
Determine whether
\[ f(z)=\overline{z} \]is complex differentiable at
\[ z=2+i. \]Practice 7: Components
For
\[ f(z)=z^2, \]identify \(u\), \(v\), \(u_x\), and \(v_x\), then recover \(f'(z)\).
Practice 8: Local Geometry
Suppose
\[ f'(z_0)=3e^{i\pi/4}. \]Describe the local first-order scaling and rotation near \(z_0\).
Practice 9: Higher Derivatives
Find
\[ f''(z) \]if
\[ f(z)=z^5. \]Practice 10: Differentiable at a Special Point
Show that
\[ f(z)=z\overline{z} \]is complex differentiable at \(z=0\), and find the derivative there.
Practice Answers
Practice 1
Start with
\[ \frac{(z+h)^2+3(z+h)-(z^2+3z)}{h}. \]Simplify the numerator:
\[ 2zh+h^2+3h. \]Divide by \(h\):
\[ 2z+h+3. \]Let \(h\to0\):
\[ \boxed{ f'(z)=2z+3. } \]Practice 2
\[ \boxed{ f'(z)=20z^3-6z^2+7. } \]Practice 3
Use the product rule:
\[ f'(z) = 3z^2e^z+z^3e^z. \]Therefore,
\[ \boxed{ f'(z)=e^z(z^3+3z^2). } \]Practice 4
Using the quotient rule,
\[ f'(z) = \frac{2z(z)-(z^2+1)}{z^2}. \]Therefore,
\[ \boxed{ f'(z) = \frac{z^2-1}{z^2} = 1-\frac1{z^2}. } \]Practice 5
By the chain rule,
\[ \boxed{ f'(z)=3e^{3z}. } \]Practice 6
The difference quotient for conjugation reduces to
\[ \frac{\overline{h}}{h}. \]It gives \(1\) along real increments and \(-1\) along imaginary increments.
Therefore,
\[ \boxed{ f\text{ is not complex differentiable at }2+i. } \]Practice 7
\[ u=x^2-y^2, \qquad v=2xy. \] \[ u_x=2x, \qquad v_x=2y. \]Therefore,
\[ f'(z) = 2x+i(2y) = \boxed{2z}. \]Practice 8
The local scale factor is
\[ \boxed{3}, \]and the local rotation is
\[ \boxed{\frac{\pi}{4}} \]counterclockwise.
Practice 9
\[ f'(z)=5z^4. \]Differentiate again:
\[ \boxed{ f''(z)=20z^3. } \]Practice 10
At \(z=0\),
\[ \frac{f(h)-f(0)}{h} = \frac{h\overline{h}}{h} = \overline{h} \]for \(h\neq0\).
As \(h\to0\),
\[ \overline{h}\to0. \]Therefore,
\[ \boxed{ f'(0)=0. } \]Checking Strategies
- When using the definition, verify that every uncanceled term has a clear limit as \(h\to0\).
- For a suspected failure of differentiability, compare real and imaginary increments first.
- If using component functions, compute the partial derivatives carefully and check their compatibility.
- For polynomial or rational functions, compare with familiar derivative rules.
- Check domain exclusions before using quotient formulas.
- Interpret \(f'(z_0)\) geometrically through its modulus and argument when that adds insight.
Connections to Earlier Mathematics
Complex differentiation connects several earlier topics:
- Limits: the derivative is fundamentally a complex limit.
- Multivariable calculus: component functions introduce partial derivatives in \(x\) and \(y\).
- Linear algebra: the derivative acts locally like a two-dimensional linear transformation.
- Polar form: the derivative's modulus and argument encode local scaling and rotation.
- Geometry: nonzero derivatives lead toward local angle preservation.
What to Remember
The complex derivative is
\[ \boxed{ f'(z) = \lim_{h\to0} \frac{f(z+h)-f(z)}{h}. } \]The crucial requirement is that the quotient approach the same value for every complex direction of \(h\to0\).
Differentiability implies continuity:
\[ \boxed{ f\text{ differentiable at }z_0 \Rightarrow f\text{ continuous at }z_0. } \]But continuity alone does not guarantee a complex derivative.
When \(f'(z_0)\neq0\), the derivative gives the local first-order transformation
\[ \boxed{ h\mapsto f'(z_0)h, } \]combining local scaling and rotation.
Where We Go Next
Writing
\[ f(z)=u(x,y)+iv(x,y) \]revealed that complex differentiability forces the real and imaginary component functions to change in tightly coordinated ways.
The next lesson, Analytic Functions, develops what happens when complex differentiability holds throughout a neighborhood or region, and explains why this stronger condition leads to extraordinary regularity. The following lesson then isolates the Cauchy–Riemann Relationships as the key component equations governing complex differentiability.