Contour Integrals of Functions of a Complex Variable Bernd Schr - PowerPoint PPT Presentation

Contours Contour Integrals Examples Definitions An arc C is a Jordan arc or a simple arc if and only if for t 1 � = t 2 we have z ( t 1 ) � = z ( t 2 ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definitions An arc C is a Jordan arc or a simple arc if and only if for t 1 � = t 2 we have z ( t 1 ) � = z ( t 2 ) . ℑ ( z ) ✻ ✲ ℜ ( z ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definitions An arc C is a Jordan arc or a simple arc if and only if for t 1 � = t 2 we have z ( t 1 ) � = z ( t 2 ) . ℑ ( z ) ✻ q ✲ ℜ ( z ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definitions logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definitions An arc C is called a simple closed curve if, except for z ( a ) = z ( b ) we have that t 1 � = t 2 implies z ( t 1 ) � = z ( t 2 ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definitions An arc C is called a simple closed curve if, except for z ( a ) = z ( b ) we have that t 1 � = t 2 implies z ( t 1 ) � = z ( t 2 ) . ℑ ( z ) ✻ ✲ ℜ ( z ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definitions An arc C is called a simple closed curve if, except for z ( a ) = z ( b ) we have that t 1 � = t 2 implies z ( t 1 ) � = z ( t 2 ) . ℑ ( z ) ✻ q ✲ ℜ ( z ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definitions An arc C is called a simple closed curve if, except for z ( a ) = z ( b ) we have that t 1 � = t 2 implies z ( t 1 ) � = z ( t 2 ) . ℑ ( z ) ✻ q ✲ ℜ ( z ) ✙ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definitions An arc C is called a simple closed curve if, except for z ( a ) = z ( b ) we have that t 1 � = t 2 implies z ( t 1 ) � = z ( t 2 ) . ℑ ( z ) ✻ q ✲ ℜ ( z ) ✙ ■ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definitions An arc C is called a simple closed curve if, except for z ( a ) = z ( b ) we have that t 1 � = t 2 implies z ( t 1 ) � = z ( t 2 ) . ℑ ( z ) ✻ q ✲ ℜ ( z ) ■ ✙ ■ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definitions An arc C is called a simple closed curve if, except for z ( a ) = z ( b ) we have that t 1 � = t 2 implies z ( t 1 ) � = z ( t 2 ) . ℑ ( z ) ✻ q ✲ ℜ ( z ) ■ ✐ ✙ ■ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definitions An arc C is called a simple closed curve if, except for z ( a ) = z ( b ) we have that t 1 � = t 2 implies z ( t 1 ) � = z ( t 2 ) . ℑ ( z ) ✻ q ✲ ② ℜ ( z ) ■ ✐ ✙ ■ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definitions An arc C is called a simple closed curve if, except for z ( a ) = z ( b ) we have that t 1 � = t 2 implies z ( t 1 ) � = z ( t 2 ) . ℑ ( z ) ✻ q ✲ ② ℜ ( z ) ✙ ■ ✐ ✙ ■ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definitions An arc C is called a simple closed curve if, except for z ( a ) = z ( b ) we have that t 1 � = t 2 implies z ( t 1 ) � = z ( t 2 ) . ℑ ( z ) ✻ q ✲ ② ℜ ( z ) ✻ ✙ ■ ✐ ✙ ■ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definitions logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definitions A simple closed curve is called positively oriented if and only if it is traversed in the counterclockwise (mathematically positive) direction. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definitions A simple closed curve is called positively oriented if and only if it is traversed in the counterclockwise (mathematically positive) direction. ℑ ( z ) ✻ ✲ ℜ ( z ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definitions A simple closed curve is called positively oriented if and only if it is traversed in the counterclockwise (mathematically positive) direction. ℑ ( z ) ✻ ❑ ✲ ℜ ( z ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definitions logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definitions An arc C is called smooth if and only if the function z ( t ) that traverses C is differentiable with continuous derivative and z ′ ( t ) � = 0 for all t . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definitions An arc C is called smooth if and only if the function z ( t ) that traverses C is differentiable with continuous derivative and z ′ ( t ) � = 0 for all t . ℑ ( z ) ✻ ❃ ✶ C ❄ ❃ ✲ � x ( a ) , y ( a ) � ❯ r ❯ ℜ ( z ) ⑥ ② � � x ( b ) , y ( b ) r logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definitions An arc C is called smooth if and only if the function z ( t ) that traverses C is differentiable with continuous derivative and z ′ ( t ) � = 0 for all t . ℑ ( z ) ✻ ❃ ✶ C ❄ ❃ � r ✲ � x ( a ) , y ( a ) ❯ ❯ ℜ ( z ) ⑥ � � x ( b ) , y ( b ) r logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definitions An arc C is called smooth if and only if the function z ( t ) that traverses C is differentiable with continuous derivative and z ′ ( t ) � = 0 for all t . ℑ ( z ) ✻ ❃ ✶ C ❄ ❃ � r ✲ � x ( a ) , y ( a ) ❯ ❯ ℜ ( z ) ⑥ � � x ( b ) , y ( b ) r No! logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definitions logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definitions An arc is called a contour or a piecewise smooth arc if and only if it consists of smooth arcs joined end-to-end. It is called a simple closed contour if and only if there is no self-intersection except that the initial point equals the final point. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definitions An arc is called a contour or a piecewise smooth arc if and only if it consists of smooth arcs joined end-to-end. It is called a simple closed contour if and only if there is no self-intersection except that the initial point equals the final point. ℑ ( z ) ✻ ❃ ✶ C ❄ ❃ � r ✲ � x ( a ) , y ( a ) ❯ ❯ ℜ ( z ) ⑥ � x ( b ) , y ( b ) � r logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definitions An arc is called a contour or a piecewise smooth arc if and only if it consists of smooth arcs joined end-to-end. It is called a simple closed contour if and only if there is no self-intersection except that the initial point equals the final point. ℑ ( z ) ✻ ❃ ✶ C ❄ ❃ � r ✲ � x ( a ) , y ( a ) ❯ ❯ ℜ ( z ) ⑥ � x ( b ) , y ( b ) � r O.k. for contours. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Example. With z ( θ ) = e i θ and 0 ≤ θ ≤ 2 π , the unit circle C = { z ∈ C : | z | = 1 } is a positively oriented simple closed curve. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Example. With z ( θ ) = e i θ and 0 ≤ θ ≤ 2 π , the unit circle C = { z ∈ C : | z | = 1 } is a positively oriented simple closed curve. ℑ ( z ) ✻ ✲ ℜ ( z ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Example. With z ( θ ) = e i θ and 0 ≤ θ ≤ 2 π , the unit circle C = { z ∈ C : | z | = 1 } is a positively oriented simple closed curve. ℑ ( z ) ✻ ❑ ✲ ℜ ( z ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Example. With z ( θ ) = e − i θ and 0 ≤ θ ≤ 2 π , the unit circle C = { z ∈ C : | z | = 1 } is a simple closed curve, but it is not positively oriented. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Example. With z ( θ ) = e − i θ and 0 ≤ θ ≤ 2 π , the unit circle C = { z ∈ C : | z | = 1 } is a simple closed curve, but it is not positively oriented. ℑ ( z ) ✻ ✲ ℜ ( z ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Example. With z ( θ ) = e − i θ and 0 ≤ θ ≤ 2 π , the unit circle C = { z ∈ C : | z | = 1 } is a simple closed curve, but it is not positively oriented. ℑ ( z ) ✻ ❯ ✲ ℜ ( z ) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definition. The length of a contour C parametrized by z ( t ) is � b � z ′ ( t ) � � � dt . L : = a logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definition. The length of a contour C parametrized by z ( t ) is � b � z ′ ( t ) � � � dt . L : = a Discussion. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definition. The length of a contour C parametrized by z ( t ) is � b � z ′ ( t ) � � � dt . L : = a Discussion. � b � z ′ ( t ) � � dt � L = a logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definition. The length of a contour C parametrized by z ( t ) is � b � z ′ ( t ) � � � dt . L : = a Discussion. � b � z ′ ( t ) � � dt � L = a � b �� � 2 + � 2 dt x ′ ( t ) � y ′ ( t ) = a logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definition. The length of a contour C parametrized by z ( t ) is � b � z ′ ( t ) � � � dt . L : = a Discussion. � b � z ′ ( t ) � � � dt L = a � b �� � 2 + � 2 dt x ′ ( t ) � y ′ ( t ) = a which is the length formula from multivariable calculus. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definition. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definition. Let f be a continuous function of a complex variable and let C be a contour with parametrization z ( t ) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definition. Let f be a continuous function of a complex variable and let C be a contour with parametrization z ( t ) . Then we define the contour integral of f over C as � C f ( z ) dz logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Definition. Let f be a continuous function of a complex variable and let C be a contour with parametrization z ( t ) . Then we define the contour integral of f over C as � b � a f ( z ( t )) z ′ ( t ) dt . C f ( z ) dz : = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Note. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Note. If φ is a differentiable function with continuous, nonzero derivative that maps the interval [ α , β ] to the interval [ a , b ] , then z ( φ ( τ )) is another parametrization of C . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Note. If φ is a differentiable function with continuous, nonzero derivative that maps the interval [ α , β ] to the interval [ a , b ] , then z ( φ ( τ )) is another parametrization of C . It just uses the parameter τ in [ α , β ] rather than the parameter t in [ a , b ] . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Note. If φ is a differentiable function with continuous, nonzero derivative that maps the interval [ α , β ] to the interval [ a , b ] , then z ( φ ( τ )) is another parametrization of C . It just uses the parameter τ in [ α , β ] rather than the parameter t in [ a , b ] . But the integral of f over C is not affected by interchanging parametrizations in this fashion, because with ξ ( τ ) : = z ( φ ( τ )) we have logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Note. If φ is a differentiable function with continuous, nonzero derivative that maps the interval [ α , β ] to the interval [ a , b ] , then z ( φ ( τ )) is another parametrization of C . It just uses the parameter τ in [ α , β ] rather than the parameter t in [ a , b ] . But the integral of f over C is not affected by interchanging parametrizations in this fashion, because with ξ ( τ ) : = z ( φ ( τ )) we have � β α f ( ξ ( τ )) ξ ′ ( τ ) d τ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Note. If φ is a differentiable function with continuous, nonzero derivative that maps the interval [ α , β ] to the interval [ a , b ] , then z ( φ ( τ )) is another parametrization of C . It just uses the parameter τ in [ α , β ] rather than the parameter t in [ a , b ] . But the integral of f over C is not affected by interchanging parametrizations in this fashion, because with ξ ( τ ) : = z ( φ ( τ )) we have � β � β α f ( ξ ( τ )) ξ ′ ( τ ) d τ α f ( z ( φ ( τ ))) z ′ ( φ ( τ )) φ ′ ( τ ) d τ = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Note. If φ is a differentiable function with continuous, nonzero derivative that maps the interval [ α , β ] to the interval [ a , b ] , then z ( φ ( τ )) is another parametrization of C . It just uses the parameter τ in [ α , β ] rather than the parameter t in [ a , b ] . But the integral of f over C is not affected by interchanging parametrizations in this fashion, because with ξ ( τ ) : = z ( φ ( τ )) we have � β � β α f ( ξ ( τ )) ξ ′ ( τ ) d τ α f ( z ( φ ( τ ))) z ′ ( φ ( τ )) φ ′ ( τ ) d τ = � b a f ( z ( t )) z ′ ( t ) dt = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Note. If φ is a differentiable function with continuous, nonzero derivative that maps the interval [ α , β ] to the interval [ a , b ] , then z ( φ ( τ )) is another parametrization of C . It just uses the parameter τ in [ α , β ] rather than the parameter t in [ a , b ] . But the integral of f over C is not affected by interchanging parametrizations in this fashion, because with ξ ( τ ) : = z ( φ ( τ )) we have � β � β α f ( ξ ( τ )) ξ ′ ( τ ) d τ α f ( z ( φ ( τ ))) z ′ ( φ ( τ )) φ ′ ( τ ) d τ = � b a f ( z ( t )) z ′ ( t ) dt = Therefore, the definition of the contour integral is sensible, as it only depends on the shape of the contour, not on the way we parametrize it. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Note. If φ is a differentiable function with continuous, nonzero derivative that maps the interval [ α , β ] to the interval [ a , b ] , then z ( φ ( τ )) is another parametrization of C . It just uses the parameter τ in [ α , β ] rather than the parameter t in [ a , b ] . But the integral of f over C is not affected by interchanging parametrizations in this fashion, because with ξ ( τ ) : = z ( φ ( τ )) we have � β � β α f ( ξ ( τ )) ξ ′ ( τ ) d τ α f ( z ( φ ( τ ))) z ′ ( φ ( τ )) φ ′ ( τ ) d τ = � b a f ( z ( t )) z ′ ( t ) dt = Therefore, the definition of the contour integral is sensible, as it only depends on the shape of the contour, not on the way we parametrize it. (Omitted proof that any two parametrizations “differ” by a φ as above.) logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Rules. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Rules. � � C wf ( z ) dz = w C f ( z ) dz . 1. If w is a complex number, then logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Rules. � � C wf ( z ) dz = w C f ( z ) dz . 1. If w is a complex number, then � � � 2. C ( f + g )( z ) dz = C f ( z ) dz + C g ( z ) dz . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Rules. � � C wf ( z ) dz = w C f ( z ) dz . 1. If w is a complex number, then � � � 2. C ( f + g )( z ) dz = C f ( z ) dz + C g ( z ) dz . 3. Let − C denote the same contour as C , only traversed in the opposite direction. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Rules. � � C wf ( z ) dz = w C f ( z ) dz . 1. If w is a complex number, then � � � 2. C ( f + g )( z ) dz = C f ( z ) dz + C g ( z ) dz . 3. Let − C denote the same contour as C , only traversed in the opposite direction. Then z ( − τ ) with − b ≤ τ ≤ − a is a parametrization and logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Rules. � � C wf ( z ) dz = w C f ( z ) dz . 1. If w is a complex number, then � � � 2. C ( f + g )( z ) dz = C f ( z ) dz + C g ( z ) dz . 3. Let − C denote the same contour as C , only traversed in the opposite direction. Then z ( − τ ) with − b ≤ τ ≤ − a is a parametrization and � − C f ( z ) dz logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Rules. � � C wf ( z ) dz = w C f ( z ) dz . 1. If w is a complex number, then � � � 2. C ( f + g )( z ) dz = C f ( z ) dz + C g ( z ) dz . 3. Let − C denote the same contour as C , only traversed in the opposite direction. Then z ( − τ ) with − b ≤ τ ≤ − a is a parametrization and � − a � − b f ( z ( − τ )) d − C f ( z ) dz = d τ z ( − τ ) d τ logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Rules. � � C wf ( z ) dz = w C f ( z ) dz . 1. If w is a complex number, then � � � 2. C ( f + g )( z ) dz = C f ( z ) dz + C g ( z ) dz . 3. Let − C denote the same contour as C , only traversed in the opposite direction. Then z ( − τ ) with − b ≤ τ ≤ − a is a parametrization and � − a � − a � − b f ( z ( − τ )) d − b f ( z ( − τ )) z ′ ( − τ )( − 1 ) d τ − C f ( z ) dz = d τ z ( − τ ) d τ = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Rules. � � C wf ( z ) dz = w C f ( z ) dz . 1. If w is a complex number, then � � � 2. C ( f + g )( z ) dz = C f ( z ) dz + C g ( z ) dz . 3. Let − C denote the same contour as C , only traversed in the opposite direction. Then z ( − τ ) with − b ≤ τ ≤ − a is a parametrization and � − a � − a � − b f ( z ( − τ )) d − b f ( z ( − τ )) z ′ ( − τ )( − 1 ) d τ − C f ( z ) dz = d τ z ( − τ ) d τ = � a b f ( z ( t )) z ′ ( t ) dt = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Rules. � � C wf ( z ) dz = w C f ( z ) dz . 1. If w is a complex number, then � � � 2. C ( f + g )( z ) dz = C f ( z ) dz + C g ( z ) dz . 3. Let − C denote the same contour as C , only traversed in the opposite direction. Then z ( − τ ) with − b ≤ τ ≤ − a is a parametrization and � − a � − a � − b f ( z ( − τ )) d − b f ( z ( − τ )) z ′ ( − τ )( − 1 ) d τ − C f ( z ) dz = d τ z ( − τ ) d τ = � a � b f ( z ( t )) z ′ ( t ) dt = − = C f ( z ) dz logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Rules. � � C wf ( z ) dz = w C f ( z ) dz . 1. If w is a complex number, then � � � 2. C ( f + g )( z ) dz = C f ( z ) dz + C g ( z ) dz . 3. Let − C denote the same contour as C , only traversed in the opposite direction. Then z ( − τ ) with − b ≤ τ ≤ − a is a parametrization and � − a � − a � − b f ( z ( − τ )) d − b f ( z ( − τ )) z ′ ( − τ )( − 1 ) d τ − C f ( z ) dz = d τ z ( − τ ) d τ = � a � b f ( z ( t )) z ′ ( t ) dt = − = C f ( z ) dz 4. If the endpoint of C 1 is the starting point of C 2 , then the union of the two contours in denoted C : = C 1 + C 2 and we have � � � C f ( z ) dz = f ( z ) dz + g ( z ) dz . C 1 C 2 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Example. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Example. Compute the contour integral of f ( z ) = | z | around the upper half of the positively oriented unit circle. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Example. Compute the contour integral of f ( z ) = | z | around the upper half of the positively oriented unit circle. � C f ( z ) dz logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Example. Compute the contour integral of f ( z ) = | z | around the upper half of the positively oriented unit circle. � π � � � e it � C f ( z ) dz = � 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Example. Compute the contour integral of f ( z ) = | z | around the upper half of the positively oriented unit circle. � π � � ie it dt � � e it � C f ( z ) dz = 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Example. Compute the contour integral of f ( z ) = | z | around the upper half of the positively oriented unit circle. � π � � ie it dt � � e it � C f ( z ) dz = 0 � π 0 ie it dt = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Example. Compute the contour integral of f ( z ) = | z | around the upper half of the positively oriented unit circle. � π � � ie it dt � e it � � C f ( z ) dz = 0 � π 0 ie it dt = π e it � = � � 0 logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Example. Compute the contour integral of f ( z ) = | z | around the upper half of the positively oriented unit circle. � π � � ie it dt � e it � � C f ( z ) dz = 0 � π 0 ie it dt = π e it � = � � 0 e i π − e 0 = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Example. Compute the contour integral of f ( z ) = | z | around the upper half of the positively oriented unit circle. � π � � ie it dt � e it � � C f ( z ) dz = 0 � π 0 ie it dt = π e it � = � � 0 e i π − e 0 = − 2 = logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Theorem. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Theorem. If the complex function f ( z ) has an antiderivative F ( z ) , then the integral of f over the contour C parametrized with z ( t ) , a ≤ t ≤ b is equal to � C f ( z ) dz = F ( z ( b )) − F ( z ( a )) . logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable

Contours Contour Integrals Examples Theorem. If the complex function f ( z ) has an antiderivative F ( z ) , then the integral of f over the contour C parametrized with z ( t ) , a ≤ t ≤ b is equal to � C f ( z ) dz = F ( z ( b )) − F ( z ( a )) . Proof. logo1 Bernd Schr¨ oder Louisiana Tech University, College of Engineering and Science Contour Integrals of Functions of a Complex Variable