Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Definite Integrals of Complex Valued
Functions
Bernd Schröder
Bernd Schröder
Definite Integrals of Complex Valued Functions
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Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Introduction
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Introduction
1. Ultimately, we want to integrate complex valued functions
along paths (“contours”) in the complex plane.
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Introduction
1. Ultimately, we want to integrate complex valued functions
along paths (“contours”) in the complex plane.
2. To avoid working with Riemann sums and less-than-handy
definitions, we will define these integrals as integrals of
complex valued functions of a real variable.
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Analyzing Functions That Map Real Numbers to
Complex Numbers
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Analyzing Functions That Map Real Numbers to
Complex Numbers
1. A function w(t) that maps real numbers to complex
numbers has a real part u(t) and an imaginary part v(t).
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Analyzing Functions That Map Real Numbers to
Complex Numbers
1. A function w(t) that maps real numbers to complex
numbers has a real part u(t) and an imaginary part v(t).
2. That is, w(t) = u(t) + iv(t).
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Analyzing Functions That Map Real Numbers to
Complex Numbers
1. A function w(t) that maps real numbers to complex
numbers has a real part u(t) and an imaginary part v(t).
2. That is, w(t) = u(t) + iv(t).
3. If u and v are differentiable, then the derivative of w is
w0 (t) := u0 (t) + iv0 (t).
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Analyzing Functions That Map Real Numbers to
Complex Numbers
1. A function w(t) that maps real numbers to complex
numbers has a real part u(t) and an imaginary part v(t).
2. That is, w(t) = u(t) + iv(t).
3. If u and v are differentiable, then the derivative of w is
w0 (t) := u0 (t) + iv0 (t).
4. This is exactly the same as for derivatives of vector valued
functions
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Analyzing Functions That Map Real Numbers to
Complex Numbers
1. A function w(t) that maps real numbers to complex
numbers has a real part u(t) and an imaginary part v(t).
2. That is, w(t) = u(t) + iv(t).
3. If u and v are differentiable, then the derivative of w is
w0 (t) := u0 (t) + iv0 (t).
4. This is exactly the same as for derivatives of vector valued
functions: Derivatives are taken componentwise.
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Analyzing Functions That Map Real Numbers to
Complex Numbers
1. A function w(t) that maps real numbers to complex
numbers has a real part u(t) and an imaginary part v(t).
2. That is, w(t) = u(t) + iv(t).
3. If u and v are differentiable, then the derivative of w is
w0 (t) := u0 (t) + iv0 (t).
4. This is exactly the same as for derivatives of vector valued
functions: Derivatives are taken componentwise.
5. We could formally define the derivative using difference
quotients.
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Analyzing Functions That Map Real Numbers to
Complex Numbers
1. A function w(t) that maps real numbers to complex
numbers has a real part u(t) and an imaginary part v(t).
2. That is, w(t) = u(t) + iv(t).
3. If u and v are differentiable, then the derivative of w is
w0 (t) := u0 (t) + iv0 (t).
4. This is exactly the same as for derivatives of vector valued
functions: Derivatives are taken componentwise.
5. We could formally define the derivative using difference
quotients.
6. But all we would do would be to prove that derivatives are
taken componentwise and then use that fact over and over.
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Analyzing Functions That Map Real Numbers to
Complex Numbers
1. A function w(t) that maps real numbers to complex
numbers has a real part u(t) and an imaginary part v(t).
2. That is, w(t) = u(t) + iv(t).
3. If u and v are differentiable, then the derivative of w is
w0 (t) := u0 (t) + iv0 (t).
4. This is exactly the same as for derivatives of vector valued
functions: Derivatives are taken componentwise.
5. We could formally define the derivative using difference
quotients.
6. But all we would do would be to prove that derivatives are
taken componentwise and then use that fact over and over.
So we use the componentwise approach as the definition.
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Familiar Differentiation Formulas Carry Over
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Familiar Differentiation Formulas Carry Over
Let λ = a + ib ∈ C.
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Familiar Differentiation Formulas Carry Over
Let λ = a + ib ∈ C. Then
d λt
e
dt
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Familiar Differentiation Formulas Carry Over
Let λ = a + ib ∈ C. Then
d λt
d (a+ib)t
=
e
e
dt
dt
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Familiar Differentiation Formulas Carry Over
Let λ = a + ib ∈ C. Then
d λt
d (a+ib)t
d h at ibt i
=
=
e
e
e e
dt
dt
dt
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Familiar Differentiation Formulas Carry Over
Let λ = a + ib ∈ C. Then
d λt
d (a+ib)t
d h at ibt i
d at
=
=
=
e
e
e e
e (cos(bt) + i sin(bt))
dt
dt
dt
dt
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Familiar Differentiation Formulas Carry Over
Let λ = a + ib ∈ C. Then
d λt
d (a+ib)t
d h at ibt i
d at
=
=
=
e
e
e e
e (cos(bt) + i sin(bt))
dt
dt
dt
dt
d at
d at
=
e cos(bt) + i
e sin(bt)
dt
dt
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Familiar Differentiation Formulas Carry Over
Let λ = a + ib ∈ C. Then
d λt
d (a+ib)t
d h at ibt i
d at
=
=
=
e
e
e e
e (cos(bt) + i sin(bt))
dt
dt
dt
dt
d at
d at
=
e cos(bt) + i
e sin(bt)
dt
dt
= aeat cos(bt) − beat sin(bt)
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Familiar Differentiation Formulas Carry Over
Let λ = a + ib ∈ C. Then
d λt
d (a+ib)t
d h at ibt i
d at
=
=
=
e
e
e e
e (cos(bt) + i sin(bt))
dt
dt
dt
dt
d at
d at
=
e cos(bt) + i
e sin(bt)
dt
dt
= aeat cos(bt) − beat sin(bt) + i aeat sin(bt) + beat cos(bt)
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Familiar Differentiation Formulas Carry Over
Let λ = a + ib ∈ C. Then
d λt
d (a+ib)t
d h at ibt i
d at
=
=
=
e
e
e e
e (cos(bt) + i sin(bt))
dt
dt
dt
dt
d at
d at
=
e cos(bt) + i
e sin(bt)
dt
dt
= aeat cos(bt) − beat sin(bt) + i aeat sin(bt) + beat cos(bt)
= aeat cos(bt) + i sin(bt) + beat − sin(bt) + i cos(bt)
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Familiar Differentiation Formulas Carry Over
Let λ = a + ib ∈ C. Then
d λt
d (a+ib)t
d h at ibt i
d at
=
=
=
e
e
e e
e (cos(bt) + i sin(bt))
dt
dt
dt
dt
d at
d at
=
e cos(bt) + i
e sin(bt)
dt
dt
= aeat cos(bt) − beat sin(bt) + i aeat sin(bt) + beat cos(bt)
= aeat cos(bt) + i sin(bt) + beat − sin(bt) + i cos(bt)
= aeat eibt + ibeat i sin(bt) + cos(bt)
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Familiar Differentiation Formulas Carry Over
Let λ = a + ib ∈ C. Then
d λt
d (a+ib)t
d h at ibt i
d at
=
=
=
e
e
e e
e (cos(bt) + i sin(bt))
dt
dt
dt
dt
d at
d at
=
e cos(bt) + i
e sin(bt)
dt
dt
= aeat cos(bt) − beat sin(bt) + i aeat sin(bt) + beat cos(bt)
= aeat cos(bt) + i sin(bt) + beat − sin(bt) + i cos(bt)
= aeat eibt + ibeat i sin(bt) + cos(bt)
= aeat eibt + ibeat eibt
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Familiar Differentiation Formulas Carry Over
Let λ = a + ib ∈ C. Then
d λt
d (a+ib)t
d h at ibt i
d at
=
=
=
e
e
e e
e (cos(bt) + i sin(bt))
dt
dt
dt
dt
d at
d at
=
e cos(bt) + i
e sin(bt)
dt
dt
= aeat cos(bt) − beat sin(bt) + i aeat sin(bt) + beat cos(bt)
= aeat cos(bt) + i sin(bt) + beat − sin(bt) + i cos(bt)
= aeat eibt + ibeat i sin(bt) + cos(bt)
= aeat eibt + ibeat eibt = (a + ib)eat eibt
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Familiar Differentiation Formulas Carry Over
Let λ = a + ib ∈ C. Then
d λt
d (a+ib)t
d h at ibt i
d at
=
=
=
e
e
e e
e (cos(bt) + i sin(bt))
dt
dt
dt
dt
d at
d at
=
e cos(bt) + i
e sin(bt)
dt
dt
= aeat cos(bt) − beat sin(bt) + i aeat sin(bt) + beat cos(bt)
= aeat cos(bt) + i sin(bt) + beat − sin(bt) + i cos(bt)
= aeat eibt + ibeat i sin(bt) + cos(bt)
= aeat eibt + ibeat eibt = (a + ib)eat eibt = λ eλ t .
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Definite Integrals are Defined Componentwise,
Too
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Definite Integrals are Defined Componentwise,
Too
For w(t) = u(t) + iv(t) defined for all t ∈ [a, b] we define
Z b
w(t) dt
a
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Definite Integrals are Defined Componentwise,
Too
For w(t) = u(t) + iv(t) defined for all t ∈ [a, b] we define
Z b
Z b
u(t) dt + i
w(t) dt :=
a
Bernd Schröder
Definite Integrals of Complex Valued Functions
Z b
a
v(t) dt
a
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Definite Integrals are Defined Componentwise,
Too
For w(t) = u(t) + iv(t) defined for all t ∈ [a, b] we define
Z b
Z b
u(t) dt + i
w(t) dt :=
a
Z b
a
v(t) dt
a
provided that the integrals of u and v exist.
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Example.
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Z π
cos(t) + i sin(t) dt.
Example. Compute the integral
0
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Z π
cos(t) + i sin(t) dt.
Example. Compute the integral
0
Z π
cos(t) + i sin(t) dt
0
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Z π
cos(t) + i sin(t) dt.
Example. Compute the integral
0
Z π
Z π
cos(t) + i sin(t) dt =
0
Bernd Schröder
Definite Integrals of Complex Valued Functions
Z π
cos(t) dt + i
0
sin(t) dt
0
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Z π
cos(t) + i sin(t) dt.
Example. Compute the integral
0
Z π
Z π
Z π
cos(t) + i sin(t) dt =
0
Bernd Schröder
Definite Integrals of Complex Valued Functions
cos(t) dt + i
sin(t) dt
0
π
π
= sin(t) 0 − i cos(t) 0
0
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Z π
cos(t) + i sin(t) dt.
Example. Compute the integral
0
Z π
Z π
Z π
cos(t) + i sin(t) dt =
0
Bernd Schröder
Definite Integrals of Complex Valued Functions
cos(t) dt + i
sin(t) dt
0
π
π
= sin(t) 0 − i cos(t) 0
= 0 − i(−2)
0
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Z π
cos(t) + i sin(t) dt.
Example. Compute the integral
0
Z π
Z π
Z π
cos(t) + i sin(t) dt =
0
Bernd Schröder
Definite Integrals of Complex Valued Functions
cos(t) dt + i
sin(t) dt
0
π
π
= sin(t) 0 − i cos(t) 0
= 0 − i(−2) = 2i
0
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
The Fundamental Theorem of Calculus Holds,
Too
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
The Fundamental Theorem of Calculus Holds,
Too
If w(t) = u(t) + iv(t) has an antiderivative W(t) = U(t) + iV(t),
then
Z
b
w(t) dt
a
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
The Fundamental Theorem of Calculus Holds,
Too
If w(t) = u(t) + iv(t) has an antiderivative W(t) = U(t) + iV(t),
then
Z
b
a
w(t) dt = W(t)|ba
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
The Fundamental Theorem of Calculus Holds,
Too
If w(t) = u(t) + iv(t) has an antiderivative W(t) = U(t) + iV(t),
then
Z
b
a
w(t) dt = W(t)|ba = W(b) − W(a).
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
The Fundamental Theorem of Calculus Holds,
Too
If w(t) = u(t) + iv(t) has an antiderivative W(t) = U(t) + iV(t),
then
Z
b
a
w(t) dt = W(t)|ba = W(b) − W(a).
Proof.
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
The Fundamental Theorem of Calculus Holds,
Too
If w(t) = u(t) + iv(t) has an antiderivative W(t) = U(t) + iV(t),
then
Z
b
a
w(t) dt = W(t)|ba = W(b) − W(a).
Proof. Note that
u(t) + iv(t)
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
The Fundamental Theorem of Calculus Holds,
Too
If w(t) = u(t) + iv(t) has an antiderivative W(t) = U(t) + iV(t),
then
Z
b
a
w(t) dt = W(t)|ba = W(b) − W(a).
Proof. Note that
u(t) + iv(t) = w(t)
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
The Fundamental Theorem of Calculus Holds,
Too
If w(t) = u(t) + iv(t) has an antiderivative W(t) = U(t) + iV(t),
then
Z
b
a
w(t) dt = W(t)|ba = W(b) − W(a).
Proof. Note that
u(t) + iv(t) = w(t) = W 0 (t)
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
The Fundamental Theorem of Calculus Holds,
Too
If w(t) = u(t) + iv(t) has an antiderivative W(t) = U(t) + iV(t),
then
Z
b
a
w(t) dt = W(t)|ba = W(b) − W(a).
Proof. Note that
u(t) + iv(t) = w(t) = W 0 (t)
= U 0 (t) + iV 0 (t)
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
The Fundamental Theorem of Calculus Holds,
Too
If w(t) = u(t) + iv(t) has an antiderivative W(t) = U(t) + iV(t),
then
Z
b
a
w(t) dt = W(t)|ba = W(b) − W(a).
Proof. Note that
u(t) + iv(t) = w(t) = W 0 (t)
= U 0 (t) + iV 0 (t)
So U 0 = u and V 0 = v and hence
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
The Fundamental Theorem of Calculus Holds,
Too
If w(t) = u(t) + iv(t) has an antiderivative W(t) = U(t) + iV(t),
then
Z
b
a
w(t) dt = W(t)|ba = W(b) − W(a).
Proof. Note that
u(t) + iv(t) = w(t) = W 0 (t)
= U 0 (t) + iV 0 (t)
So U 0 = u and V 0 = v and hence
Z b
w(t) dt
a
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
The Fundamental Theorem of Calculus Holds,
Too
If w(t) = u(t) + iv(t) has an antiderivative W(t) = U(t) + iV(t),
then
Z
b
a
w(t) dt = W(t)|ba = W(b) − W(a).
Proof. Note that
u(t) + iv(t) = w(t) = W 0 (t)
= U 0 (t) + iV 0 (t)
So U 0 = u and V 0 = v and hence
Z b
Z b
w(t) dt =
a
Z b
u(t) dt + i
a
Bernd Schröder
Definite Integrals of Complex Valued Functions
v(t) dt
a
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
The Fundamental Theorem of Calculus Holds,
Too
If w(t) = u(t) + iv(t) has an antiderivative W(t) = U(t) + iV(t),
then
Z
b
a
w(t) dt = W(t)|ba = W(b) − W(a).
Proof. Note that
u(t) + iv(t) = w(t) = W 0 (t)
= U 0 (t) + iV 0 (t)
So U 0 = u and V 0 = v and hence
Z b
Z b
w(t) dt =
a
Z b
u(t) dt + i
a
Bernd Schröder
Definite Integrals of Complex Valued Functions
a
v(t) dt = U(t)|ba + V(t)|ba
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
The Fundamental Theorem of Calculus Holds,
Too
If w(t) = u(t) + iv(t) has an antiderivative W(t) = U(t) + iV(t),
then
Z
b
a
w(t) dt = W(t)|ba = W(b) − W(a).
Proof. Note that
u(t) + iv(t) = w(t) = W 0 (t)
= U 0 (t) + iV 0 (t)
So U 0 = u and V 0 = v and hence
Z b
Z b
w(t) dt =
a
Z b
u(t) dt + i
a
a
v(t) dt = U(t)|ba + V(t)|ba
= W(t)|ba
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
The Fundamental Theorem of Calculus Holds,
Too
If w(t) = u(t) + iv(t) has an antiderivative W(t) = U(t) + iV(t),
then
Z
b
a
w(t) dt = W(t)|ba = W(b) − W(a).
Proof. Note that
u(t) + iv(t) = w(t) = W 0 (t)
= U 0 (t) + iV 0 (t)
So U 0 = u and V 0 = v and hence
Z b
Z b
w(t) dt =
a
Z b
u(t) dt + i
a
a
v(t) dt = U(t)|ba + V(t)|ba
= W(t)|ba
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Example.
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Z π
Example. Compute the integral
eit dt.
0
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Z π
Example. Compute the integral
eit dt.
0
Z π
eit dt
0
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Z π
Example. Compute the integral
eit dt.
0
Z π
0
1 it π
e dt =
e i 0
it
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Z π
Example. Compute the integral
eit dt.
0
Z π
0
1 it π
e dt =
e i 0
= −ieiπ − (−i)e0
it
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science
Differentiating Complex Valued Functions of a Real Variable
Integrating Complex Valued Functions of a Real Variable
Z π
Example. Compute the integral
eit dt.
0
Z π
0
1 it π
e dt =
e i 0
= −ieiπ − (−i)e0 = 2i
it
Bernd Schröder
Definite Integrals of Complex Valued Functions
logo1
Louisiana Tech University, College of Engineering and Science