Derivatives
Differentiation Formulas
Derivatives of Complex Functions
Bernd Schröder
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Introduction
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Introduction
1. The idea for the derivative lies in the desire to compute
instantaneous velocities or slopes of tangent lines.
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Introduction
1. The idea for the derivative lies in the desire to compute
instantaneous velocities or slopes of tangent lines.
2. In both cases, we want to know what happens when the
f (x + h) − f (x)
denominator in the difference quotient
goes to
h
zero.
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Introduction
1. The idea for the derivative lies in the desire to compute
instantaneous velocities or slopes of tangent lines.
2. In both cases, we want to know what happens when the
f (x + h) − f (x)
denominator in the difference quotient
goes to
h
zero.
3. Although the visualization is challenging
Bernd Schröder
Derivatives of Complex Functions
logo1
Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Introduction
1. The idea for the derivative lies in the desire to compute
instantaneous velocities or slopes of tangent lines.
2. In both cases, we want to know what happens when the
f (x + h) − f (x)
denominator in the difference quotient
goes to
h
zero.
3. Although the visualization is challenging, if not impossible
Bernd Schröder
Derivatives of Complex Functions
logo1
Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Introduction
1. The idea for the derivative lies in the desire to compute
instantaneous velocities or slopes of tangent lines.
2. In both cases, we want to know what happens when the
f (x + h) − f (x)
denominator in the difference quotient
goes to
h
zero.
3. Although the visualization is challenging, if not impossible, we
can investigate the same question for functions that map complex
numbers to complex numbers.
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Introduction
1. The idea for the derivative lies in the desire to compute
instantaneous velocities or slopes of tangent lines.
2. In both cases, we want to know what happens when the
f (x + h) − f (x)
denominator in the difference quotient
goes to
h
zero.
3. Although the visualization is challenging, if not impossible, we
can investigate the same question for functions that map complex
numbers to complex numbers.
4. After all, the algebra and the idea of a limit translate to C.
Bernd Schröder
Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Definition.
Bernd Schröder
Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Definition. Let f be defined in a neighborhood of the point z0 .
Bernd Schröder
Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Definition. Let f be defined in a neighborhood of the point z0 . Then f
is called differentiable at z0
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Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Definition. Let f be defined in a neighborhood of the point z0 . Then f
is called differentiable at z0 if and only if the limit
lim
z→z0
f (z) − f (z0 )
z − z0
exists.
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Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Definition. Let f be defined in a neighborhood of the point z0 . Then f
is called differentiable at z0 if and only if the limit
lim
z→z0
f (z) − f (z0 )
z − z0
exists.
In this case we set
f (z) − f (z0 )
z→z0
z − z0
f 0 (z0 ) := lim
Bernd Schröder
Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Definition. Let f be defined in a neighborhood of the point z0 . Then f
is called differentiable at z0 if and only if the limit
lim
z→z0
f (z) − f (z0 )
z − z0
exists.
In this case we set
f (z) − f (z0 )
z→z0
z − z0
and call it the derivative of f at z0 .
f 0 (z0 ) := lim
Bernd Schröder
Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Definition. Let f be defined in a neighborhood of the point z0 . Then f
is called differentiable at z0 if and only if the limit
lim
z→z0
f (z) − f (z0 )
z − z0
exists.
In this case we set
f (z) − f (z0 )
z→z0
z − z0
and call it the derivative of f at z0 . Other notations for the derivative
df
at z0 are (z0 )
dz
f 0 (z0 ) := lim
Bernd Schröder
Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Definition. Let f be defined in a neighborhood of the point z0 . Then f
is called differentiable at z0 if and only if the limit
lim
z→z0
f (z) − f (z0 )
z − z0
exists.
In this case we set
f (z) − f (z0 )
z→z0
z − z0
and call it the derivative of f at z0 . Other notations for the derivative
df
at z0 are (z0 ) and Df (z0 ).
dz
f 0 (z0 ) := lim
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Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
The function f with an open domain is called differentiable
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Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
The function f with an open domain is called differentiable (or
holomorphic or analytic)
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Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
The function f with an open domain is called differentiable (or
holomorphic or analytic) if and only if it is differentiable at every z0
in its domain.
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Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
The function f with an open domain is called differentiable (or
holomorphic or analytic) if and only if it is differentiable at every z0
in its domain. The function f is called entire if and only if it is
differentiable at every z ∈ C.
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Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
The function f with an open domain is called differentiable (or
holomorphic or analytic) if and only if it is differentiable at every z0
in its domain. The function f is called entire if and only if it is
differentiable at every z ∈ C. If f is not differentiable at z0 , but for
every neighborhood of z0 there is a point so that f is analytic in a
neighborhood around that point, then z0 is called a singularity.
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
The function f with an open domain is called differentiable (or
holomorphic or analytic) if and only if it is differentiable at every z0
in its domain. The function f is called entire if and only if it is
differentiable at every z ∈ C. If f is not differentiable at z0 , but for
every neighborhood of z0 there is a point so that f is analytic in a
neighborhood around that point, then z0 is called a singularity.
It can be proved that f is differentiable if and only if
f (z0 + ∆z) − f (z0 )
lim
exists.
∆z→0
∆z
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
The function f with an open domain is called differentiable (or
holomorphic or analytic) if and only if it is differentiable at every z0
in its domain. The function f is called entire if and only if it is
differentiable at every z ∈ C. If f is not differentiable at z0 , but for
every neighborhood of z0 there is a point so that f is analytic in a
neighborhood around that point, then z0 is called a singularity.
It can be proved that f is differentiable if and only if
f (z0 + ∆z) − f (z0 )
lim
exists. In this case, the above limit is the
∆z→0
∆z
derivative.
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
The function f with an open domain is called differentiable (or
holomorphic or analytic) if and only if it is differentiable at every z0
in its domain. The function f is called entire if and only if it is
differentiable at every z ∈ C. If f is not differentiable at z0 , but for
every neighborhood of z0 there is a point so that f is analytic in a
neighborhood around that point, then z0 is called a singularity.
It can be proved that f is differentiable if and only if
f (z0 + ∆z) − f (z0 )
lim
exists. In this case, the above limit is the
∆z→0
∆z
derivative.
With ∆w := f (z0 + ∆z) − f (z0 ) we also write
Bernd Schröder
Derivatives of Complex Functions
dw
∆w
= lim
.
dz ∆z→0 ∆z
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Derivatives
Differentiation Formulas
Example.
Bernd Schröder
Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Example. Compute the derivative of f (z) = z3 .
Bernd Schröder
Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Example. Compute the derivative of f (z) = z3 .
dz3
dz
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Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Example. Compute the derivative of f (z) = z3 .
dz3
dz
Bernd Schröder
Derivatives of Complex Functions
=
(z + ∆z)3 − z3
∆z→0
∆z
lim
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Derivatives
Differentiation Formulas
Example. Compute the derivative of f (z) = z3 .
dz3
dz
Bernd Schröder
Derivatives of Complex Functions
(z + ∆z)3 − z3
∆z→0
∆z
3
z + 3z2 ∆z + 3z∆z2 + ∆z3 − z3
= lim
∆z→0
∆z
=
lim
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Derivatives
Differentiation Formulas
Example. Compute the derivative of f (z) = z3 .
dz3
dz
Bernd Schröder
Derivatives of Complex Functions
(z + ∆z)3 − z3
∆z→0
∆z
3
z + 3z2 ∆z + 3z∆z2 + ∆z3 − z3
= lim
∆z→0
∆z
2
2
3z ∆z + 3z∆z + ∆z3
= lim
∆z→0
∆z
=
lim
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Derivatives
Differentiation Formulas
Example. Compute the derivative of f (z) = z3 .
dz3
dz
Bernd Schröder
Derivatives of Complex Functions
(z + ∆z)3 − z3
∆z→0
∆z
3
z + 3z2 ∆z + 3z∆z2 + ∆z3 − z3
= lim
∆z→0
∆z
2
2
3z ∆z + 3z∆z + ∆z3
= lim
∆z→0
∆z
2
∆z 3z + 3z∆z + ∆z2
= lim
∆z→0
∆z
=
lim
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Derivatives
Differentiation Formulas
Example. Compute the derivative of f (z) = z3 .
dz3
dz
=
=
=
=
=
(z + ∆z)3 − z3
∆z→0
∆z
3
z + 3z2 ∆z + 3z∆z2 + ∆z3 − z3
lim
∆z→0
∆z
2
2
3z ∆z + 3z∆z + ∆z3
lim
∆z→0
∆z
2
∆z 3z + 3z∆z + ∆z2
lim
∆z→0
∆z
2
lim 3z + 3z∆z + ∆z2
lim
∆z→0
Bernd Schröder
Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Example. Compute the derivative of f (z) = z3 .
dz3
dz
=
=
=
=
=
(z + ∆z)3 − z3
∆z→0
∆z
3
z + 3z2 ∆z + 3z∆z2 + ∆z3 − z3
lim
∆z→0
∆z
2
2
3z ∆z + 3z∆z + ∆z3
lim
∆z→0
∆z
2
∆z 3z + 3z∆z + ∆z2
lim
∆z→0
∆z
2
lim 3z + 3z∆z + ∆z2
lim
∆z→0
2
= 3z
Bernd Schröder
Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Example.
Bernd Schröder
Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Example. Show that f (z) = z is not differentiable at any z0 ∈ C.
Bernd Schröder
Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Example. Show that f (z) = z is not differentiable at any z0 ∈ C.
lim
∆z→0
Bernd Schröder
Derivatives of Complex Functions
z0 + ∆z − z0
∆z
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Derivatives
Differentiation Formulas
Example. Show that f (z) = z is not differentiable at any z0 ∈ C.
lim
∆z→0
Bernd Schröder
Derivatives of Complex Functions
z0 + ∆z − z0
∆z
=
lim
∆z→0
z0 + ∆z − z0
∆z
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Derivatives
Differentiation Formulas
Example. Show that f (z) = z is not differentiable at any z0 ∈ C.
lim
∆z→0
Bernd Schröder
Derivatives of Complex Functions
z0 + ∆z − z0
∆z
z0 + ∆z − z0
∆z
∆z
= lim
∆z→0 ∆z
=
lim
∆z→0
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Derivatives
Differentiation Formulas
Example. Show that f (z) = z is not differentiable at any z0 ∈ C.
lim
∆z→0
z0 + ∆z − z0
∆z
z0 + ∆z − z0
∆z
∆z
= lim
∆z→0 ∆z
=
lim
∆z→0
... which does not exist.
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Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Theorem.
Bernd Schröder
Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Theorem. If the complex function f is differentiable at z0 , then f is
continuous at z0 .
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Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Theorem. If the complex function f is differentiable at z0 , then f is
continuous at z0 . Hence, every differentiable function is continuous.
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Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Theorem. If the complex function f is differentiable at z0 , then f is
continuous at z0 . Hence, every differentiable function is continuous.
However, not every continuous function is differentiable.
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Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Theorem. If the complex function f is differentiable at z0 , then f is
continuous at z0 . Hence, every differentiable function is continuous.
However, not every continuous function is differentiable.
Proof.
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Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Theorem. If the complex function f is differentiable at z0 , then f is
continuous at z0 . Hence, every differentiable function is continuous.
However, not every continuous function is differentiable.
Proof. For continuity at z0 note that
0
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Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Theorem. If the complex function f is differentiable at z0 , then f is
continuous at z0 . Hence, every differentiable function is continuous.
However, not every continuous function is differentiable.
Proof. For continuity at z0 note that
0 =
lim (z − z0 ) lim
z→z0
Bernd Schröder
Derivatives of Complex Functions
z→z0
f (z) − f (z0 )
z − z0
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Derivatives
Differentiation Formulas
Theorem. If the complex function f is differentiable at z0 , then f is
continuous at z0 . Hence, every differentiable function is continuous.
However, not every continuous function is differentiable.
Proof. For continuity at z0 note that
0 =
lim (z − z0 ) lim
z→z0
Bernd Schröder
Derivatives of Complex Functions
z→z0
f (z) − f (z0 )
f (z) − f (z0 )
= lim (z − z0 )
z→z
z − z0
z − z0
0
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Derivatives
Differentiation Formulas
Theorem. If the complex function f is differentiable at z0 , then f is
continuous at z0 . Hence, every differentiable function is continuous.
However, not every continuous function is differentiable.
Proof. For continuity at z0 note that
0 =
=
f (z) − f (z0 )
f (z) − f (z0 )
= lim (z − z0 )
z→z
z − z0
z − z0
0
lim f (z) − f (z0 ).
lim (z − z0 ) lim
z→z0
z→z0
z→z0
Bernd Schröder
Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Theorem. If the complex function f is differentiable at z0 , then f is
continuous at z0 . Hence, every differentiable function is continuous.
However, not every continuous function is differentiable.
Proof. For continuity at z0 note that
0 =
=
f (z) − f (z0 )
f (z) − f (z0 )
= lim (z − z0 )
z→z
z − z0
z − z0
0
lim f (z) − f (z0 ).
lim (z − z0 ) lim
z→z0
z→z0
z→z0
For the last statement, consider f (z) = z.
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Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Theorem. If the complex function f is differentiable at z0 , then f is
continuous at z0 . Hence, every differentiable function is continuous.
However, not every continuous function is differentiable.
Proof. For continuity at z0 note that
0 =
=
f (z) − f (z0 )
f (z) − f (z0 )
= lim (z − z0 )
z→z
z − z0
z − z0
0
lim f (z) − f (z0 ).
lim (z − z0 ) lim
z→z0
z→z0
z→z0
For the last statement, consider f (z) = z.
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Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Theorem.
Bernd Schröder
Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Theorem. Let f and g be differentiable at z0 and let c ∈ C.
Bernd Schröder
Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Theorem. Let f and g be differentiable at z0 and let c ∈ C. Then the
functions f + g, f − g and cf are all differentiable at z0
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Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Theorem. Let f and g be differentiable at z0 and let c ∈ C. Then the
functions f + g, f − g and cf are all differentiable at z0 and
(f + g)0 (z0 ) = f 0 (z0 ) + g0 (z0 ),
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Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Theorem. Let f and g be differentiable at z0 and let c ∈ C. Then the
functions f + g, f − g and cf are all differentiable at z0 and
(f + g)0 (z0 ) = f 0 (z0 ) + g0 (z0 ),
(f − g)0 (z0 ) = f 0 (z0 ) − g0 (z0 ),
Bernd Schröder
Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Theorem. Let f and g be differentiable at z0 and let c ∈ C. Then the
functions f + g, f − g and cf are all differentiable at z0 and
(f + g)0 (z0 ) = f 0 (z0 ) + g0 (z0 ),
(f − g)0 (z0 ) = f 0 (z0 ) − g0 (z0 ),
(cf )0 (z0 ) = cf 0 (z0 ).
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Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Proof (addition only).
Bernd Schröder
Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Proof (addition only).
(f + g)0 (z0 )
Bernd Schröder
Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Proof (addition only).
(f + g)0 (z0 )
(f + g)(z) − (f + g)(z0 )
= lim
z→z0
z − z0
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Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Proof (addition only).
(f + g)0 (z0 )
(f + g)(z) − (f + g)(z0 )
f (z) + g(z) − f (z0 ) − g(z0 )
= lim
= lim
z→z0
z→z0
z − z0
z − z0
Bernd Schröder
Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Proof (addition only).
(f + g)0 (z0 )
(f + g)(z) − (f + g)(z0 )
f (z) + g(z) − f (z0 ) − g(z0 )
= lim
= lim
z→z0
z→z0
z − z0
z − z0
f (z) − f (z0 ) + g(z) − g(z0 )
= lim
z→z0
z − z0
Bernd Schröder
Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Proof (addition only).
(f + g)0 (z0 )
(f + g)(z) − (f + g)(z0 )
f (z) + g(z) − f (z0 ) − g(z0 )
= lim
= lim
z→z0
z→z0
z − z0
z − z0
f (z) − f (z0 ) + g(z) − g(z0 )
= lim
z→z0
z − z0
f (z) − f (z0 )
g(z) − g(z0 )
= lim
+ lim
z→z0
z→z
z − z0
z − z0
0
Bernd Schröder
Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Proof (addition only).
(f + g)0 (z0 )
(f + g)(z) − (f + g)(z0 )
f (z) + g(z) − f (z0 ) − g(z0 )
= lim
= lim
z→z0
z→z0
z − z0
z − z0
f (z) − f (z0 ) + g(z) − g(z0 )
= lim
z→z0
z − z0
f (z) − f (z0 )
g(z) − g(z0 )
= lim
+ lim
z→z0
z→z
z − z0
z − z0
0
= f 0 (z0 ) + g0 (z0 )
Bernd Schröder
Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Proof (addition only).
(f + g)0 (z0 )
(f + g)(z) − (f + g)(z0 )
f (z) + g(z) − f (z0 ) − g(z0 )
= lim
= lim
z→z0
z→z0
z − z0
z − z0
f (z) − f (z0 ) + g(z) − g(z0 )
= lim
z→z0
z − z0
f (z) − f (z0 )
g(z) − g(z0 )
= lim
+ lim
z→z0
z→z
z − z0
z − z0
0
= f 0 (z0 ) + g0 (z0 )
Bernd Schröder
Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Theorem.
Bernd Schröder
Derivatives of Complex Functions
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Derivatives
Differentiation Formulas
Theorem. Product and Quotient Rule. Let f and g be differentiable
at z0 .
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Theorem. Product and Quotient Rule. Let f and g be differentiable
at z0 . Then fg is differentiable at x
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Theorem. Product and Quotient Rule. Let f and g be differentiable
at z0 . Then fg is differentiable at x with
(fg)0 (z0 ) = f 0 (z0 )g(z0 ) + g0 (z0 )f (z0 ).
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Theorem. Product and Quotient Rule. Let f and g be differentiable
at z0 . Then fg is differentiable at x with
(fg)0 (z0 ) = f 0 (z0 )g(z0 ) + g0 (z0 )f (z0 ).
f
Moreover, if g(z0 ) 6= 0, then the quotient is differentiable at z0
g
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Theorem. Product and Quotient Rule. Let f and g be differentiable
at z0 . Then fg is differentiable at x with
(fg)0 (z0 ) = f 0 (z0 )g(z0 ) + g0 (z0 )f (z0 ).
f
Moreover, if g(z0 ) 6= 0, then the quotient is differentiable at z0 with
g
0
0
f
f (z0 )g(z0 ) − g0 (z0 )f (z0 )
.
(z0 ) =
2
g
g(z0 )
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Proof (quotient rule only).
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Proof (quotient rule only).
lim
f
f
g (z) − g (z0 )
z→z0
z − z0
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Proof (quotient rule only).
lim
f
f
g (z) − g (z0 )
z − z0
z→z0
=
lim
z→z0
f (z)
g(z)
f (z0 )
− g(z
0)
z − z0
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Proof (quotient rule only).
lim
f
f
g (z) − g (z0 )
z − z0
z→z0
=
lim
z→z0
f (z)
g(z)
f (z0 )
− g(z
0)
z − z0
Bernd Schröder
Derivatives of Complex Functions
= lim
z→z0
f (z)g(z0 )−f (z0 )g(z)
g(z)g(z0 )
z − z0
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Proof (quotient rule only).
lim
f
f
g (z) − g (z0 )
z − z0
z→z0
=
f (z)
g(z)
f (z0 )
− g(z
0)
f (z)g(z0 )−f (z0 )g(z)
g(z)g(z0 )
= lim
z→z0
z − z0
z − z0
1
f (z)g(z0 ) − f (z0 )g(z)
= lim
z→z0 g(z)g(z0 )
z − z0
lim
z→z0
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Proof (quotient rule only).
lim
f
f
g (z) − g (z0 )
z − z0
z→z0
=
f (z)
g(z)
f (z0 )
− g(z
0)
f (z)g(z0 )−f (z0 )g(z)
g(z)g(z0 )
= lim
z→z0
z − z0
z − z0
1
f (z)g(z0 ) − f (z0 )g(z)
= lim
z→z0 g(z)g(z0 )
z − z0
1
f (z)g(z0 ) − f (z0 )g(z0 ) + f (z0 )g(z0 ) − f (z0 )g(z)
= lim
z→z0 g(z)g(z0 )
z − z0
lim
z→z0
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Proof (quotient rule only).
lim
f
f
g (z) − g (z0 )
z − z0
z→z0
=
f (z)
g(z)
f (z0 )
− g(z
0)
f (z)g(z0 )−f (z0 )g(z)
g(z)g(z0 )
= lim
z→z0
z − z0
z − z0
1
f (z)g(z0 ) − f (z0 )g(z)
= lim
z→z0 g(z)g(z0 )
z − z0
1
f (z)g(z0 ) − f (z0 )g(z0 ) + f (z0 )g(z0 ) − f (z0 )g(z)
= lim
z→z0 g(z)g(z0 )
z − z0
f (z)g(z0 ) − f (z0 )g(z0 ) − f (z0 )g(z) − f (z0 )g(z0 )
1
= lim
z→z0 g(z)g(z0 )
z − z0
lim
z→z0
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Proof (quotient rule only).
lim
f
f
g (z) − g (z0 )
z − z0
z→z0
=
=
=
=
=
f (z)
g(z)
f (z0 )
− g(z
0)
f (z)g(z0 )−f (z0 )g(z)
g(z)g(z0 )
= lim
z→z0
z − z0
z − z0
1
f (z)g(z0 ) − f (z0 )g(z)
lim
z→z0 g(z)g(z0 )
z − z0
1
f (z)g(z0 ) − f (z0 )g(z0 ) + f (z0 )g(z0 ) − f (z0 )g(z)
lim
z→z0 g(z)g(z0 )
z − z0
f (z)g(z0 ) − f (z0 )g(z0 ) − f (z0 )g(z) − f (z0 )g(z0 )
1
lim
z→z0 g(z)g(z0 )
z − z0
1
f (z)g(z0 ) − f (z0 )g(z0 ) f (z0 )g(z) − f (z0 )g(z0 )
lim
−
z→z0 g(z)g(z0 )
z − z0
z − z0
lim
z→z0
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Proof (quotient rule cont.).
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Proof (quotient rule cont.).
1
f (z)g(z0 ) − f (z0 )g(z0 ) f (z0 )g(z) − f (z0 )g(z0 )
lim
−
z→z0 g(z)g(z0 )
z − z0
z − z0
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Proof (quotient rule cont.).
1
f (z)g(z0 ) − f (z0 )g(z0 ) f (z0 )g(z) − f (z0 )g(z0 )
lim
−
z→z0 g(z)g(z0 )
z − z0
z − z0
f (z) − f (z0 )
g(z) − g(z0 )
1
g(z0 ) −
f (z0 )
= lim
z→z0 g(z)g(z0 )
z − z0
z − z0
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Proof (quotient rule cont.).
1
f (z)g(z0 ) − f (z0 )g(z0 ) f (z0 )g(z) − f (z0 )g(z0 )
lim
−
z→z0 g(z)g(z0 )
z − z0
z − z0
f (z) − f (z0 )
g(z) − g(z0 )
1
g(z0 ) −
f (z0 )
= lim
z→z0 g(z)g(z0 )
z − z0
z − z0
1
0
0
=
2 f (z0 )g(z0 ) − g (z0 )f (z0 )
g(z0 )
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Proof (quotient rule cont.).
1
f (z)g(z0 ) − f (z0 )g(z0 ) f (z0 )g(z) − f (z0 )g(z0 )
lim
−
z→z0 g(z)g(z0 )
z − z0
z − z0
f (z) − f (z0 )
g(z) − g(z0 )
1
g(z0 ) −
f (z0 )
= lim
z→z0 g(z)g(z0 )
z − z0
z − z0
1
0
0
=
2 f (z0 )g(z0 ) − g (z0 )f (z0 )
g(z0 )
0
f (z0 )g(z0 ) − g0 (z0 )f (z0 )
=
.
2
g(z0 )
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Proof (quotient rule cont.).
1
f (z)g(z0 ) − f (z0 )g(z0 ) f (z0 )g(z) − f (z0 )g(z0 )
lim
−
z→z0 g(z)g(z0 )
z − z0
z − z0
f (z) − f (z0 )
g(z) − g(z0 )
1
g(z0 ) −
f (z0 )
= lim
z→z0 g(z)g(z0 )
z − z0
z − z0
1
0
0
=
2 f (z0 )g(z0 ) − g (z0 )f (z0 )
g(z0 )
0
f (z0 )g(z0 ) − g0 (z0 )f (z0 )
=
.
2
g(z0 )
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Theorem.
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Theorem. Chain Rule.
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Theorem. Chain Rule. Let f , g be complex functions and let z0 be
such that g is differentiable at z0 and f is differentiable at g(z0 ).
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Theorem. Chain Rule. Let f , g be complex functions and let z0 be
such that g is differentiable at z0 and f is differentiable at g(z0 ). Then
f ◦ g is differentiable at z0
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Theorem. Chain Rule. Let f , g be complex functions and let z0 be
such that g is differentiable at z0 and f is differentiable at g(z0 ). Then
f ◦ g is differentiable at
z0 and the derivative is
(f ◦ g)0 (z0 ) = f 0 g(z0 ) g0 (z0 ).
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Proof (small technicality ignored).
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Proof (small technicality ignored).
f ◦ g(z) − f ◦ g(z0 )
z→z0
z − z0
lim
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Proof (small technicality ignored).
f ◦ g(z) − f ◦ g(z0 )
z→z0
z − z0
f g(z) − f g(z0 ) g(z) − g(z0 )
= lim
·
z→z0
z − z0
g(z) − g(z0 )
lim
Bernd Schröder
Derivatives of Complex Functions
logo1
Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Proof (small technicality ignored).
f ◦ g(z) − f ◦ g(z0 )
z→z0
z − z0
f g(z) − f g(z0 ) g(z) − g(z0 )
= lim
·
z→z0
z − z0
g(z) − g(z0 )
f g(z) − f g(z0 ) g(z) − g(z0 )
= lim
·
z→z0
g(z) − g(z0 )
z − z0
lim
Bernd Schröder
Derivatives of Complex Functions
logo1
Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Proof (small technicality ignored).
f ◦ g(z) − f ◦ g(z0 )
z→z0
z − z0
f g(z) − f g(z0 ) g(z) − g(z0 )
= lim
·
z→z0
z − z0
g(z) − g(z0 )
f g(z) − f g(z0 ) g(z) − g(z0 )
= lim
·
z→z0
g(z) − g(z0 )
z − z0
f g(z) − f g(z0 )
g(z) − g(z0 )
= lim
· lim
z→z0
z→z0
g(z) − g(z0 )
z − z0
lim
Bernd Schröder
Derivatives of Complex Functions
logo1
Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Proof (small technicality ignored).
f ◦ g(z) − f ◦ g(z0 )
z→z0
z − z0
f g(z) − f g(z0 ) g(z) − g(z0 )
= lim
·
z→z0
z − z0
g(z) − g(z0 )
f g(z) − f g(z0 ) g(z) − g(z0 )
= lim
·
z→z0
g(z) − g(z0 )
z − z0
f g(z) − f g(z0 )
g(z) − g(z0 )
= lim
· lim
z→z0
z→z0
g(z) − g(z0 )
z − z0
f (u) − f g(z0 )
g(z) − g(z0 )
=
lim
· lim
z→z0
u − g(z0 )
z − z0
u→g(z0 )
lim
Bernd Schröder
Derivatives of Complex Functions
logo1
Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Proof (small technicality ignored).
f ◦ g(z) − f ◦ g(z0 )
z→z0
z − z0
f g(z) − f g(z0 ) g(z) − g(z0 )
= lim
·
z→z0
z − z0
g(z) − g(z0 )
f g(z) − f g(z0 ) g(z) − g(z0 )
= lim
·
z→z0
g(z) − g(z0 )
z − z0
f g(z) − f g(z0 )
g(z) − g(z0 )
= lim
· lim
z→z0
z→z0
g(z) − g(z0 )
z − z0
f (u) − f g(z0 )
g(z) − g(z0 )
=
lim
· lim
z→z0
u − g(z0 )
z − z0
u→g(z0 )
0
0
= f g(z0 ) g (z0 ).
lim
Bernd Schröder
Derivatives of Complex Functions
logo1
Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Proof (small technicality ignored).
f ◦ g(z) − f ◦ g(z0 )
z→z0
z − z0
f g(z) − f g(z0 ) g(z) − g(z0 )
= lim
·
z→z0
z − z0
g(z) − g(z0 )
f g(z) − f g(z0 ) g(z) − g(z0 )
= lim
·
z→z0
g(z) − g(z0 )
z − z0
f g(z) − f g(z0 )
g(z) − g(z0 )
= lim
· lim
z→z0
z→z0
g(z) − g(z0 )
z − z0
f (u) − f g(z0 )
g(z) − g(z0 )
=
lim
· lim
z→z0
u − g(z0 )
z − z0
u→g(z0 )
0
0
= f g(z0 ) g (z0 ).
lim
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Example
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Example (derivatives work just like they did in calculus).
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Example (derivatives work just like they did in calculus). Find the
2
derivative of f (x) = (iz + 4)3 z2 + 1 .
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Example (derivatives work just like they did in calculus). Find the
2
derivative of f (x) = (iz + 4)3 z2 + 1 .
2
d
(iz + 4)3 z2 + 1
dz
Bernd Schröder
Derivatives of Complex Functions
logo1
Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Example (derivatives work just like they did in calculus). Find the
2
derivative of f (x) = (iz + 4)3 z2 + 1 .
2
d
(iz + 4)3 z2 + 1
dz
2
= 3 (iz + 4)2 i z2 + 1
Bernd Schröder
Derivatives of Complex Functions
logo1
Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Example (derivatives work just like they did in calculus). Find the
2
derivative of f (x) = (iz + 4)3 z2 + 1 .
2
d
(iz + 4)3 z2 + 1
dz
2
= 3 (iz + 4)2 i z2 + 1 + (iz + 4)3 2 z2 + 1 2z
Bernd Schröder
Derivatives of Complex Functions
logo1
Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Example (derivatives work just like they did in calculus). Find the
2
derivative of f (x) = (iz + 4)3 z2 + 1 .
2
d
(iz + 4)3 z2 + 1
dz
2
= 3 (iz + 4)2 i z2 + 1 + (iz + 4)3 2 z2 + 1 2z
2
= 3i (iz + 4)2 z2 + 1 + 4z (iz + 4)3 z2 + 1
Bernd Schröder
Derivatives of Complex Functions
logo1
Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Example (derivatives work just like they did in calculus). Find the
2
derivative of f (x) = (iz + 4)3 z2 + 1 .
2
d
(iz + 4)3 z2 + 1
dz
2
= 3 (iz + 4)2 i z2 + 1 + (iz + 4)3 2 z2 + 1 2z
2
= 3i (iz + 4)2 z2 + 1 + 4z (iz + 4)3 z2 + 1
= (iz + 4)2 z2 + 1 3iz2 + 3i + 4iz2 + 16z
Bernd Schröder
Derivatives of Complex Functions
logo1
Louisiana Tech University, College of Engineering and Science
Derivatives
Differentiation Formulas
Example (derivatives work just like they did in calculus). Find the
2
derivative of f (x) = (iz + 4)3 z2 + 1 .
2
d
(iz + 4)3 z2 + 1
dz
2
= 3 (iz + 4)2 i z2 + 1 + (iz + 4)3 2 z2 + 1 2z
2
= 3i (iz + 4)2 z2 + 1 + 4z (iz + 4)3 z2 + 1
= (iz + 4)2 z2 + 1 3iz2 + 3i + 4iz2 + 16z
= (iz + 4)2 z2 + 1 7iz2 + 16z + 3i
Bernd Schröder
Derivatives of Complex Functions
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Louisiana Tech University, College of Engineering and Science