logo1

Some Reasons What we will do How We Can Do It

Why Study Functions of a Complex Variable?

Bernd Schroder

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

Why Study Complex Analysis?

1. Applications. (Harmonic Functions.) Many applications are a lotdeeper than this course will go. (Spectral theory, Fourieranalysis, number theory.)

2. First or further exposure to the idea of a proof.3. Practice working with complex numbers. (Including getting

really secure in computations.)4. Practice in two-dimensional visualization. (Contour integrals.)5. Its pretty! (Differentiable functions of a complex variable have a

lot of really nice properties.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

Why Study Complex Analysis?1. Applications.

(Harmonic Functions.) Many applications are a lotdeeper than this course will go. (Spectral theory, Fourieranalysis, number theory.)

2. First or further exposure to the idea of a proof.3. Practice working with complex numbers. (Including getting

really secure in computations.)4. Practice in two-dimensional visualization. (Contour integrals.)5. Its pretty! (Differentiable functions of a complex variable have a

lot of really nice properties.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

Why Study Complex Analysis?1. Applications. (Harmonic Functions.)

Many applications are a lotdeeper than this course will go. (Spectral theory, Fourieranalysis, number theory.)

2. First or further exposure to the idea of a proof.3. Practice working with complex numbers. (Including getting

really secure in computations.)4. Practice in two-dimensional visualization. (Contour integrals.)5. Its pretty! (Differentiable functions of a complex variable have a

lot of really nice properties.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

Why Study Complex Analysis?1. Applications. (Harmonic Functions.) Many applications are a lot

deeper than this course will go.

(Spectral theory, Fourieranalysis, number theory.)

2. First or further exposure to the idea of a proof.3. Practice working with complex numbers. (Including getting

really secure in computations.)4. Practice in two-dimensional visualization. (Contour integrals.)5. Its pretty! (Differentiable functions of a complex variable have a

lot of really nice properties.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

Why Study Complex Analysis?1. Applications. (Harmonic Functions.) Many applications are a lot

deeper than this course will go. (Spectral theory, Fourieranalysis, number theory.)

2. First or further exposure to the idea of a proof.3. Practice working with complex numbers. (Including getting

really secure in computations.)4. Practice in two-dimensional visualization. (Contour integrals.)5. Its pretty! (Differentiable functions of a complex variable have a

lot of really nice properties.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

Why Study Complex Analysis?1. Applications. (Harmonic Functions.) Many applications are a lot

deeper than this course will go. (Spectral theory, Fourieranalysis, number theory.)

2. First or further exposure to the idea of a proof.

3. Practice working with complex numbers. (Including gettingreally secure in computations.)

4. Practice in two-dimensional visualization. (Contour integrals.)5. Its pretty! (Differentiable functions of a complex variable have a

lot of really nice properties.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

Why Study Complex Analysis?1. Applications. (Harmonic Functions.) Many applications are a lot

deeper than this course will go. (Spectral theory, Fourieranalysis, number theory.)

2. First or further exposure to the idea of a proof.3. Practice working with complex numbers.

(Including gettingreally secure in computations.)

4. Practice in two-dimensional visualization. (Contour integrals.)5. Its pretty! (Differentiable functions of a complex variable have a

lot of really nice properties.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

Why Study Complex Analysis?1. Applications. (Harmonic Functions.) Many applications are a lot

deeper than this course will go. (Spectral theory, Fourieranalysis, number theory.)

2. First or further exposure to the idea of a proof.3. Practice working with complex numbers. (Including getting

really secure in computations.)

4. Practice in two-dimensional visualization. (Contour integrals.)5. Its pretty! (Differentiable functions of a complex variable have a

lot of really nice properties.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

Why Study Complex Analysis?1. Applications. (Harmonic Functions.) Many applications are a lot

deeper than this course will go. (Spectral theory, Fourieranalysis, number theory.)

2. First or further exposure to the idea of a proof.3. Practice working with complex numbers. (Including getting

really secure in computations.)4. Practice in two-dimensional visualization.

(Contour integrals.)5. Its pretty! (Differentiable functions of a complex variable have a

lot of really nice properties.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

Why Study Complex Analysis?1. Applications. (Harmonic Functions.) Many applications are a lot

deeper than this course will go. (Spectral theory, Fourieranalysis, number theory.)

2. First or further exposure to the idea of a proof.3. Practice working with complex numbers. (Including getting

really secure in computations.)4. Practice in two-dimensional visualization. (Contour integrals.)

5. Its pretty! (Differentiable functions of a complex variable have alot of really nice properties.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

Why Study Complex Analysis?1. Applications. (Harmonic Functions.) Many applications are a lot

deeper than this course will go. (Spectral theory, Fourieranalysis, number theory.)

2. First or further exposure to the idea of a proof.3. Practice working with complex numbers. (Including getting

really secure in computations.)4. Practice in two-dimensional visualization. (Contour integrals.)5. Its pretty!

(Differentiable functions of a complex variable have alot of really nice properties.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

Why Study Complex Analysis?1. Applications. (Harmonic Functions.) Many applications are a lot

deeper than this course will go. (Spectral theory, Fourieranalysis, number theory.)

2. First or further exposure to the idea of a proof.3. Practice working with complex numbers. (Including getting

really secure in computations.)4. Practice in two-dimensional visualization. (Contour integrals.)5. Its pretty! (Differentiable functions of a complex variable have a

lot of really nice properties.)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Formally Introduce Complex Numbers

1. In mathematics, only what can be proved is safe to be used.2. Learning how to do proofs takes time. So does getting used to

this axiomatic method.3. This course gives valuable (first or additional) practice, while

keeping some of the elements of a calculus-type approacharound. (Computation is good for the soul. M. Spivak)

-

6

(z)

(z)

:O

1 z1 = r1ei1

>

O2

z2 = r2ei2

7

M1 +2

z1z2 = r1r2ei(1+2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Formally Introduce Complex Numbers1. In mathematics, only what can be proved is safe to be used.

2. Learning how to do proofs takes time. So does getting used tothis axiomatic method.

3. This course gives valuable (first or additional) practice, whilekeeping some of the elements of a calculus-type approacharound. (Computation is good for the soul. M. Spivak)

-

6

(z)

(z)

:O

1 z1 = r1ei1

>

O2

z2 = r2ei2

7

M1 +2

z1z2 = r1r2ei(1+2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Formally Introduce Complex Numbers1. In mathematics, only what can be proved is safe to be used.2. Learning how to do proofs takes time.

So does getting used tothis axiomatic method.

3. This course gives valuable (first or additional) practice, whilekeeping some of the elements of a calculus-type approacharound. (Computation is good for the soul. M. Spivak)

-

6

(z)

(z)

:O

1 z1 = r1ei1

>

O2

z2 = r2ei2

7

M1 +2

z1z2 = r1r2ei(1+2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Formally Introduce Complex Numbers1. In mathematics, only what can be proved is safe to be used.2. Learning how to do proofs takes time. So does getting used to

this axiomatic method.

3. This course gives valuable (first or additional) practice, whilekeeping some of the elements of a calculus-type approacharound. (Computation is good for the soul. M. Spivak)

-

6

(z)

(z)

:O

1 z1 = r1ei1

>

O2

z2 = r2ei2

7

M1 +2

z1z2 = r1r2ei(1+2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Formally Introduce Complex Numbers1. In mathematics, only what can be proved is safe to be used.2. Learning how to do proofs takes time. So does getting used to

this axiomatic method.3. This course gives valuable (first or additional) practice

, whilekeeping some of the elements of a calculus-type approacharound. (Computation is good for the soul. M. Spivak)

-

6

(z)

(z)

:O

1 z1 = r1ei1

>

O2

z2 = r2ei2

7

M1 +2

z1z2 = r1r2ei(1+2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Formally Introduce Complex Numbers1. In mathematics, only what can be proved is safe to be used.2. Learning how to do proofs takes time. So does getting used to

this axiomatic method.3. This course gives valuable (first or additional) practice, while

keeping some of the elements of a calculus-type approacharound.

(Computation is good for the soul. M. Spivak)

-

6

(z)

(z)

:O

1 z1 = r1ei1

>

O2

z2 = r2ei2

7

M1 +2

z1z2 = r1r2ei(1+2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Formally Introduce Complex Numbers1. In mathematics, only what can be proved is safe to be used.2. Learning how to do proofs takes time. So does getting used to

this axiomatic method.3. This course gives valuable (first or additional) practice, while

keeping some of the elements of a calculus-type approacharound. (Computation is good for the soul. M. Spivak)

-

6

(z)

(z)

:O

1 z1 = r1ei1

>

O2

z2 = r2ei2

7

M1 +2

z1z2 = r1r2ei(1+2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Formally Introduce Complex Numbers1. In mathematics, only what can be proved is safe to be used.2. Learning how to do proofs takes time. So does getting used to

this axiomatic method.3. This course gives valuable (first or additional) practice, while

keeping some of the elements of a calculus-type approacharound. (Computation is good for the soul. M. Spivak)

-

6

(z)

(z)

:O

1 z1 = r1ei1

>

O2

z2 = r2ei2

7

M1 +2

z1z2 = r1r2ei(1+2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Formally Introduce Complex Numbers1. In mathematics, only what can be proved is safe to be used.2. Learning how to do proofs takes time. So does getting used to

this axiomatic method.3. This course gives valuable (first or additional) practice, while

keeping some of the elements of a calculus-type approacharound. (Computation is good for the soul. M. Spivak)

-

6

(z)

(z)

:

O1 z1 = r1ei1

>

O2

z2 = r2ei2

7

M1 +2

z1z2 = r1r2ei(1+2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Formally Introduce Complex Numbers1. In mathematics, only what can be proved is safe to be used.2. Learning how to do proofs takes time. So does getting used to

this axiomatic method.3. This course gives valuable (first or additional) practice, while

keeping some of the elements of a calculus-type approacharound. (Computation is good for the soul. M. Spivak)

-

6

(z)

(z)

:O

1 z1 = r1ei1

>

O2

z2 = r2ei2

7

M1 +2

z1z2 = r1r2ei(1+2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Formally Introduce Complex Numbers1. In mathematics, only what can be proved is safe to be used.2. Learning how to do proofs takes time. So does getting used to

this axiomatic method.3. This course gives valuable (first or additional) practice, while

keeping some of the elements of a calculus-type approacharound. (Computation is good for the soul. M. Spivak)

-

6

(z)

(z)

:O

1

z1 = r1ei1

>

O2

z2 = r2ei2

7

M1 +2

z1z2 = r1r2ei(1+2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Formally Introduce Complex Numbers1. In mathematics, only what can be proved is safe to be used.2. Learning how to do proofs takes time. So does getting used to

this axiomatic method.3. This course gives valuable (first or additional) practice, while

keeping some of the elements of a calculus-type approacharound. (Computation is good for the soul. M. Spivak)

-

6

(z)

(z)

:O

1 z1 = r1ei1

>

O2

z2 = r2ei2

7

M1 +2

z1z2 = r1r2ei(1+2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Formally Introduce Complex Numbers1. In mathematics, only what can be proved is safe to be used.2. Learning how to do proofs takes time. So does getting used to

this axiomatic method.3. This course gives valuable (first or additional) practice, while

keeping some of the elements of a calculus-type approacharound. (Computation is good for the soul. M. Spivak)

-

6

(z)

(z)

:O

1 z1 = r1ei1

>

O2

z2 = r2ei2

7

M1 +2

z1z2 = r1r2ei(1+2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Formally Introduce Complex Numbers1. In mathematics, only what can be proved is safe to be used.2. Learning how to do proofs takes time. So does getting used to

this axiomatic method.3. This course gives valuable (first or additional) practice, while

keeping some of the elements of a calculus-type approacharound. (Computation is good for the soul. M. Spivak)

-

6

(z)

(z)

:O

1 z1 = r1ei1

>

O

2

z2 = r2ei2

7

M1 +2

z1z2 = r1r2ei(1+2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Formally Introduce Complex Numbers1. In mathematics, only what can be proved is safe to be used.2. Learning how to do proofs takes time. So does getting used to

this axiomatic method.3. This course gives valuable (first or additional) practice, while

keeping some of the elements of a calculus-type approacharound. (Computation is good for the soul. M. Spivak)

-

6

(z)

(z)

:O

1 z1 = r1ei1

>

O2

z2 = r2ei2

7

M1 +2

z1z2 = r1r2ei(1+2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Formally Introduce Complex Numbers1. In mathematics, only what can be proved is safe to be used.2. Learning how to do proofs takes time. So does getting used to

this axiomatic method.3. This course gives valuable (first or additional) practice, while

keeping some of the elements of a calculus-type approacharound. (Computation is good for the soul. M. Spivak)

-

6

(z)

(z)

:O

1 z1 = r1ei1

>

O2

z2 = r2ei2

7

M1 +2

z1z2 = r1r2ei(1+2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Formally Introduce Complex Numbers1. In mathematics, only what can be proved is safe to be used.2. Learning how to do proofs takes time. So does getting used to

this axiomatic method.3. This course gives valuable (first or additional) practice, while

keeping some of the elements of a calculus-type approacharound. (Computation is good for the soul. M. Spivak)

-

6

(z)

(z)

:O

1 z1 = r1ei1

>

O2

z2 = r2ei2

7

M1 +2

z1z2 = r1r2ei(1+2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Formally Introduce Complex Numbers1. In mathematics, only what can be proved is safe to be used.2. Learning how to do proofs takes time. So does getting used to

this axiomatic method.3. This course gives valuable (first or additional) practice, while

keeping some of the elements of a calculus-type approacharound. (Computation is good for the soul. M. Spivak)

-

6

(z)

(z)

:O

1 z1 = r1ei1

>

O2

z2 = r2ei2

7

M

1 +2

z1z2 = r1r2ei(1+2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Formally Introduce Complex Numbers1. In mathematics, only what can be proved is safe to be used.2. Learning how to do proofs takes time. So does getting used to

this axiomatic method.3. This course gives valuable (first or additional) practice, while

keeping some of the elements of a calculus-type approacharound. (Computation is good for the soul. M. Spivak)

-

6

(z)

(z)

:O

1 z1 = r1ei1

>

O2

z2 = r2ei2

7

M1 +2

z1z2 = r1r2ei(1+2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Formally Introduce Complex Numbers1. In mathematics, only what can be proved is safe to be used.2. Learning how to do proofs takes time. So does getting used to

this axiomatic method.3. This course gives valuable (first or additional) practice, while

keeping some of the elements of a calculus-type approacharound. (Computation is good for the soul. M. Spivak)

-

6

(z)

(z)

:O

1 z1 = r1ei1

>

O2

z2 = r2ei2

7

M1 +2

z1z2 = r1r2ei(1+2)

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

Warning

For many (if not most) exercises, there will be no examples to mimic.Essentially, the only example that can help with a proof is the proofitself.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

WarningFor many (if not most) exercises, there will be no examples to mimic.

Essentially, the only example that can help with a proof is the proofitself.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

WarningFor many (if not most) exercises, there will be no examples to mimic.Essentially, the only example that can help with a proof is the proofitself.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Formally Introduce Differentiable Functions

1. In particular, we will gain more practice with formal limits, butwe will very quickly move on to derivatives.

2. This reflects the historical fact that derivatives (and integrals)take center stage in calculus, with limits providing the ability toformally justify everything.

3. We will also analyze the geometric nature of many of thefunctions we consider, which is a great way to practicevisualization.

f(rei

)=

(rei

)2 = r2ei2

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Formally Introduce Differentiable Functions1. In particular, we will gain more practice with formal limits

, butwe will very quickly move on to derivatives.

2. This reflects the historical fact that derivatives (and integrals)take center stage in calculus, with limits providing the ability toformally justify everything.

3. We will also analyze the geometric nature of many of thefunctions we consider, which is a great way to practicevisualization.

f(rei

)=

(rei

)2 = r2ei2

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Formally Introduce Differentiable Functions1. In particular, we will gain more practice with formal limits, but

we will very quickly move on to derivatives.

2. This reflects the historical fact that derivatives (and integrals)take center stage in calculus, with limits providing the ability toformally justify everything.

3. We will also analyze the geometric nature of many of thefunctions we consider, which is a great way to practicevisualization.

f(rei

)=

(rei

)2 = r2ei2

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Formally Introduce Differentiable Functions1. In particular, we will gain more practice with formal limits, but

we will very quickly move on to derivatives.2. This reflects the historical fact that derivatives (and integrals)

take center stage in calculus

, with limits providing the ability toformally justify everything.

3. We will also analyze the geometric nature of many of thefunctions we consider, which is a great way to practicevisualization.

f(rei

)=

(rei

)2 = r2ei2

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Formally Introduce Differentiable Functions1. In particular, we will gain more practice with formal limits, but

we will very quickly move on to derivatives.2. This reflects the historical fact that derivatives (and integrals)

take center stage in calculus, with limits providing the ability toformally justify everything.

3. We will also analyze the geometric nature of many of thefunctions we consider, which is a great way to practicevisualization.

f(rei

)=

(rei

)2 = r2ei2

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Formally Introduce Differentiable Functions1. In particular, we will gain more practice with formal limits, but

we will very quickly move on to derivatives.2. This reflects the historical fact that derivatives (and integrals)

take center stage in calculus, with limits providing the ability toformally justify everything.

3. We will also analyze the geometric nature of many of thefunctions we consider

, which is a great way to practicevisualization.

f(rei

)=

(rei

)2 = r2ei2

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Formally Introduce Differentiable Functions1. In particular, we will gain more practice with formal limits, but

we will very quickly move on to derivatives.2. This reflects the historical fact that derivatives (and integrals)

take center stage in calculus, with limits providing the ability toformally justify everything.

3. We will also analyze the geometric nature of many of thefunctions we consider, which is a great way to practicevisualization.

f(rei

)=

(rei

)2 = r2ei2

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Formally Introduce Differentiable Functions1. In particular, we will gain more practice with formal limits, but

we will very quickly move on to derivatives.2. This reflects the historical fact that derivatives (and integrals)

take center stage in calculus, with limits providing the ability toformally justify everything.

3. We will also analyze the geometric nature of many of thefunctions we consider, which is a great way to practicevisualization.

f(rei

)=

(rei

)2 = r2ei2Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Integrate Complex Functions Along Contours inthe Plane

1. This is a big topic.2. Because of the way these contour integrals behave, complex

differentiable functions are locally equal to their Taylor series.3. A wide variety of problems, including solving a variety of

improper real integrals, can be solved with integrals in thecomplex plane.

-

6(z)

(z)6

?

-

3

qi qi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Integrate Complex Functions Along Contours inthe Plane

1. This is a big topic.

2. Because of the way these contour integrals behave, complexdifferentiable functions are locally equal to their Taylor series.

3. A wide variety of problems, including solving a variety ofimproper real integrals, can be solved with integrals in thecomplex plane.

-

6(z)

(z)6

?

-

3

qi qi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Integrate Complex Functions Along Contours inthe Plane

1. This is a big topic.2. Because of the way these contour integrals behave, complex

differentiable functions are locally equal to their Taylor series.

3. A wide variety of problems, including solving a variety ofimproper real integrals, can be solved with integrals in thecomplex plane.

-

6(z)

(z)6

?

-

3

qi qi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Integrate Complex Functions Along Contours inthe Plane

1. This is a big topic.2. Because of the way these contour integrals behave, complex

differentiable functions are locally equal to their Taylor series.3. A wide variety of problems, including solving a variety of

improper real integrals, can be solved with integrals in thecomplex plane.

-

6(z)

(z)6

?

-

3

qi qi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Integrate Complex Functions Along Contours inthe Plane

1. This is a big topic.2. Because of the way these contour integrals behave, complex

differentiable functions are locally equal to their Taylor series.3. A wide variety of problems, including solving a variety of

improper real integrals, can be solved with integrals in thecomplex plane.

-

6(z)

(z)

6

?

-

3

qi qi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Integrate Complex Functions Along Contours inthe Plane

1. This is a big topic.2. Because of the way these contour integrals behave, complex

differentiable functions are locally equal to their Taylor series.3. A wide variety of problems, including solving a variety of

improper real integrals, can be solved with integrals in thecomplex plane.

-

6(z)

(z)6

?

-

3

qi qi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Integrate Complex Functions Along Contours inthe Plane

1. This is a big topic.2. Because of the way these contour integrals behave, complex

differentiable functions are locally equal to their Taylor series.3. A wide variety of problems, including solving a variety of

improper real integrals, can be solved with integrals in thecomplex plane.

-

6(z)

(z)6

?

-

3

qi qi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Integrate Complex Functions Along Contours inthe Plane

1. This is a big topic.2. Because of the way these contour integrals behave, complex

differentiable functions are locally equal to their Taylor series.3. A wide variety of problems, including solving a variety of

improper real integrals, can be solved with integrals in thecomplex plane.

-

6(z)

(z)6

?

-

3

qi qi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Integrate Complex Functions Along Contours inthe Plane

1. This is a big topic.2. Because of the way these contour integrals behave, complex

differentiable functions are locally equal to their Taylor series.3. A wide variety of problems, including solving a variety of

improper real integrals, can be solved with integrals in thecomplex plane.

-

6(z)

(z)6

?

-

3

qi qi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Integrate Complex Functions Along Contours inthe Plane

1. This is a big topic.2. Because of the way these contour integrals behave, complex

differentiable functions are locally equal to their Taylor series.3. A wide variety of problems, including solving a variety of

improper real integrals, can be solved with integrals in thecomplex plane.

-

6(z)

(z)6

?

-

3

qi qi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Integrate Complex Functions Along Contours inthe Plane

1. This is a big topic.2. Because of the way these contour integrals behave, complex

differentiable functions are locally equal to their Taylor series.3. A wide variety of problems, including solving a variety of

improper real integrals, can be solved with integrals in thecomplex plane.

-

6(z)

(z)6

?

-

3

q

i qi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Integrate Complex Functions Along Contours inthe Plane

1. This is a big topic.2. Because of the way these contour integrals behave, complex

differentiable functions are locally equal to their Taylor series.3. A wide variety of problems, including solving a variety of

improper real integrals, can be solved with integrals in thecomplex plane.

-

6(z)

(z)6

?

-

3

qi

qi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Integrate Complex Functions Along Contours inthe Plane

1. This is a big topic.2. Because of the way these contour integrals behave, complex

differentiable functions are locally equal to their Taylor series.3. A wide variety of problems, including solving a variety of

improper real integrals, can be solved with integrals in thecomplex plane.

-

6(z)

(z)6

?

-

3

qi q

i

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Integrate Complex Functions Along Contours inthe Plane

1. This is a big topic.2. Because of the way these contour integrals behave, complex

differentiable functions are locally equal to their Taylor series.3. A wide variety of problems, including solving a variety of

improper real integrals, can be solved with integrals in thecomplex plane.

-

6(z)

(z)6

?

-

3

qi qi

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

Improper real integrals can be computed via growing contour integrals.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Prove the Fundamental Theorem of Algebra

1. Every nonconstant complex polynomial has at least one complexzero.

2. We have learned this a long time ago.3. Now well also be able to tell why it is true.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Prove the Fundamental Theorem of Algebra1. Every nonconstant complex polynomial has at least one complex

zero.

2. We have learned this a long time ago.3. Now well also be able to tell why it is true.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Prove the Fundamental Theorem of Algebra1. Every nonconstant complex polynomial has at least one complex

zero.2. We have learned this a long time ago.

3. Now well also be able to tell why it is true.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

We Will Prove the Fundamental Theorem of Algebra1. Every nonconstant complex polynomial has at least one complex

zero.2. We have learned this a long time ago.3. Now well also be able to tell why it is true.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

Tips for Listening

1. We will need to digest a lot of information2. But I cannot point to the opposite end of the board for something

that we have derived 10 minutes ago.3. One way to help the limitation of the screen is to open the slides

in another window. When there is a backward reference, stop thepresentation and go to the appropriate point in the slides todouble check the claim. Putting slides and video side-by-sidecan help with cross-checks.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

Tips for Listening1. We will need to digest a lot of information

2. But I cannot point to the opposite end of the board for somethingthat we have derived 10 minutes ago.

3. One way to help the limitation of the screen is to open the slidesin another window. When there is a backward reference, stop thepresentation and go to the appropriate point in the slides todouble check the claim. Putting slides and video side-by-sidecan help with cross-checks.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

Tips for Listening1. We will need to digest a lot of information2. But I cannot point to the opposite end of the board for something

that we have derived 10 minutes ago.

3. One way to help the limitation of the screen is to open the slidesin another window. When there is a backward reference, stop thepresentation and go to the appropriate point in the slides todouble check the claim. Putting slides and video side-by-sidecan help with cross-checks.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

Tips for Listening1. We will need to digest a lot of information2. But I cannot point to the opposite end of the board for something

that we have derived 10 minutes ago.3. One way to help the limitation of the screen is to open the slides

in another window.

When there is a backward reference, stop thepresentation and go to the appropriate point in the slides todouble check the claim. Putting slides and video side-by-sidecan help with cross-checks.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

Tips for Listening1. We will need to digest a lot of information2. But I cannot point to the opposite end of the board for something

that we have derived 10 minutes ago.3. One way to help the limitation of the screen is to open the slides

in another window. When there is a backward reference, stop thepresentation and go to the appropriate point in the slides todouble check the claim.

Putting slides and video side-by-sidecan help with cross-checks.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

Tips for Listening1. We will need to digest a lot of information2. But I cannot point to the opposite end of the board for something

that we have derived 10 minutes ago.3. One way to help the limitation of the screen is to open the slides

in another window. When there is a backward reference, stop thepresentation and go to the appropriate point in the slides todouble check the claim. Putting slides and video side-by-sidecan help with cross-checks.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

When Times Get Tough

... and its likely they will.1. Remember that you did not get to this point by taking it easy.2. Difficulties come from some changes in the approach (proofs),

which take most people a while to adjust to.3. Difficulties come from computations becoming more

complicated and being paired with visualizations. Youve madethat jump before, when you went from single variable calculus tomultivariable calculus.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

When Times Get Tough... and its likely they will.

1. Remember that you did not get to this point by taking it easy.2. Difficulties come from some changes in the approach (proofs),

which take most people a while to adjust to.3. Difficulties come from computations becoming more

complicated and being paired with visualizations. Youve madethat jump before, when you went from single variable calculus tomultivariable calculus.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

When Times Get Tough... and its likely they will.

1. Remember that you did not get to this point by taking it easy.

2. Difficulties come from some changes in the approach (proofs),which take most people a while to adjust to.

3. Difficulties come from computations becoming morecomplicated and being paired with visualizations. Youve madethat jump before, when you went from single variable calculus tomultivariable calculus.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

When Times Get Tough... and its likely they will.

1. Remember that you did not get to this point by taking it easy.2. Difficulties come from some changes in the approach

(proofs),which take most people a while to adjust to.

3. Difficulties come from computations becoming morecomplicated and being paired with visualizations. Youve madethat jump before, when you went from single variable calculus tomultivariable calculus.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

When Times Get Tough... and its likely they will.

1. Remember that you did not get to this point by taking it easy.2. Difficulties come from some changes in the approach (proofs)

,which take most people a while to adjust to.

3. Difficulties come from computations becoming morecomplicated and being paired with visualizations. Youve madethat jump before, when you went from single variable calculus tomultivariable calculus.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

When Times Get Tough... and its likely they will.

1. Remember that you did not get to this point by taking it easy.2. Difficulties come from some changes in the approach (proofs),

which take most people a while to adjust to.

3. Difficulties come from computations becoming morecomplicated and being paired with visualizations. Youve madethat jump before, when you went from single variable calculus tomultivariable calculus.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

When Times Get Tough... and its likely they will.

1. Remember that you did not get to this point by taking it easy.2. Difficulties come from some changes in the approach (proofs),

which take most people a while to adjust to.3. Difficulties come from computations becoming more

complicated and being paired with visualizations.

Youve madethat jump before, when you went from single variable calculus tomultivariable calculus.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

logo1

Some Reasons What we will do How We Can Do It

When Times Get Tough... and its likely they will.

1. Remember that you did not get to this point by taking it easy.2. Difficulties come from some changes in the approach (proofs),

which take most people a while to adjust to.3. Difficulties come from computations becoming more

complicated and being paired with visualizations. Youve madethat jump before, when you went from single variable calculus tomultivariable calculus.

Bernd Schroder Louisiana Tech University, College of Engineering and Science

Why Study Functions of a Complex Variable?

Some ReasonsWhat we will doHow We Can Do It