wi4243AP/wi4244AP: Complex Analysis week 3, Monday K. P. Hart - PowerPoint PPT Presentation

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions wi4243AP/wi4244AP: Complex Analysis week 3, Monday K. P. Hart Faculty EEMCS TU Delft Delft, 15 September, 2014 K. P. Hart wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions Outline 2.4: Cauchy-Riemann equations 1 2.5: Analyticity 2 2.6: Harmonic functions 3 3.1: The Exponential function 4 3.2: Trigonometric functions 5 3.3: Logarithmic functions 6 K. P. Hart wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions Changing variables We have x = 1 z ) and y = 1 2 ( z + ¯ 2 i ( z − ¯ z ), so f can also be considered as a function of z and ¯ z . Apply the multi-variable chain-rule: ∂ f z = ∂ x ∂ x + ∂ y ∂ f ∂ y = 1 ∂ f ∂ x − 1 ∂ f ∂ f ∂ ¯ ∂ ¯ z ∂ ¯ z 2 2 i ∂ y K. P. Hart wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions Changing variables Use real and imaginary parts and the Cauchy-Riemann equations: ∂ f z = 1 2( u x + iv x ) − 1 2 i ( u y + iv y ) = 1 2( u x − v y ) − 1 2 i ( v x + u y ) = 0 ∂ ¯ So, . . . , f is (complex) differentiable iff ∂ f z = 0 ∂ ¯ K. P. Hart wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions Analytic functions Definition A function f is analytic at z 0 if it is differentiable on some neighbourhood N ( z 0 , ε ) of z 0 . f is then also analytic at all points of N ( z 0 , ε ) the domain of an analytic function is open f ( z ) = | z | 2 is differentiable at 0 but not analytic K. P. Hart wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions Orthogonal curves Useful fact: if f = u + iv is analytic then the level curves u ( x , y ) = α and v ( x , y ) = β are always orthogonal. Use implicit differentiation d y / d x = − u x / u y on level curves of u d y / d x = − v x / v y on level curves of v Apply Cauchy-Riemann equations: � d y � d y � � � � � � − u x − v x = − u x · u y · = · = − 1 d x d x u y v y u y u x u v So there. K. P. Hart wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions Example: z 2 Because z 2 = x 2 − y 2 + 2 xyi we have the level curves of u = x 2 − y 2 (red) v = 2 xy (blue) . K. P. Hart wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions Harmonic functions Definition A function φ : R 2 → R is harmonic on a domain D if it is twice differentiable and φ xx ( x , y ) + φ yy ( x , y ) = 0 on all of D If f = u + iv is analytic then u and v are harmonic. u xx = v yx and v xx = − u yx u yy = − v xy and v yy = u xy Now add. We will see later that all these derivatives actually exist K. P. Hart wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions Harmonic conjugate If u and v are harmonic and such that f = u + iv is analytic then v is a harmonic conjugate of u . Equivalently: u and v satisfy u x = v y and u y = − v x — the Cauchy-Riemann equations. This is not symmetric: u is not a conjugate of v . z 2 = x 2 − y 2 + 2 xyi is analytic but iz 2 = 2 xy + ( x 2 − y 2 ) i is not. K. P. Hart wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions Finding harmonic conjugates If u is harmonic is there a harmonic conjugate? On simply connected domains: yes. How to find it? Force the Cauchy-Riemann equations to hold: v can be written as � � v ( x , y ) = − u y d x and v ( x , y ) = u x d y Just try this! K. P. Hart wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions 2 ln( x 2 + y 2 ) Example: 1 2 ln( x 2 + y 2 ) is harmonic on the right-hand half plane u ( x , y ) = 1 (check) We have x y u x = x 2 + y 2 and u y = x 2 + y 2 Integrate � y � x v ( x , y ) = − x 2 + y 2 d x and v ( x , y ) = x 2 + y 2 d y K. P. Hart wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions 2 ln( x 2 + y 2 ) Example: 1 We get � x 2 + y 2 d y = arctan y x v ( x , y ) = x + h 1 ( x ) and � x 2 + y 2 d x = − arctan x y v ( x , y ) = − y + h 2 ( y ) Remember: if x > 0 then arctan x + arctan 1 x = π 2 and if x < 0 then arctan x + arctan 1 x = − π 2 K. P. Hart wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions 2 ln( x 2 + y 2 ) Example: 1 Since arctan y x and − arctan x y differ by a constant we conclude that v ( x , y ) = arctan y x + c on the right-hand half plane. K. P. Hart wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions 2 ln( x 2 + y 2 ) Example: 1 2 ln( x 2 + y 2 ) is harmonic on the whole complex plane, Note: 1 except at (0 , 0); can we define v ( x , y ) everywhere too? Suppose we want v (1 , 1) = π 4 We must choose v ( x , y ) = arctan y x on the right-hand half plane On the upper half plane we (must) take v ( x , y ) = − arctan x y + π 2 On the lower half plane we must take v ( x , y ) = − arctan x y − π 2 (because v (1 , − 1) = − π 4 ) But now we cannot get past the negative real axis. So: simple connectivity is necessary. K. P. Hart wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions We look at e z Remember Definition If z = x + iy then, by definition, e z = e x (cos y + i sin y ) Re e z = e x cos y and Im e z = e x sin y | e z | = e x and arg e z = y e z e w = e z + w e z +2 π i = e z K. P. Hart wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions e z is an entire function We have seen: e z is real differentiable everywhere and � u x � e x cos y − e x sin y � � u y = e x sin y e x cos y v x v y so, by the C-R equations, it is complex differentiable everywhere. It is an entire function (analytic on the whole complex plane). ( e z ) ′ = e z : the matrix on the right represents multiplication by e z . K. P. Hart wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions It is the only sensible choice Theorem f ( z ) = e z is the only function that satisfies 1 it is entire 2 f ′ ( z ) = f ( z ) 3 f (0) = 1 See the book for a derivation. K. P. Hart wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions Mapping behaviour Level curves of e x cos y (red) and e x sin y (blue) K. P. Hart wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions sin z and cos z Definition We use the Euler formulas to define sin z = e iz − e − iz and cos z = e iz + e − iz 2 i 2 Compare with hyperbolic functions: sin z = 1 i sinh iz and cos z = cosh iz or sinh z = 1 i sin iz and cosh z = cos iz K. P. Hart wi4243AP/wi4244AP: Complex Analysis

2.4: Cauchy-Riemann equations 2.5: Analyticity 2.6: Harmonic functions 3.1: The Exponential function 3.2: Trigonometric functions 3.3: Logarithmic functions Formulas Use addition formulas: sin( x + iy ) = sin x cos iy + cos x sin iy = sin x cosh y + i cos x sinh y and cos( x + iy ) = cos x cos iy − sin x sin iy = cos x cosh y − i sin x sinh y After some manipulations: � � sin 2 x + sinh 2 y and | cos z | = cos 2 x + sinh 2 y | sin z | = K. P. Hart wi4243AP/wi4244AP: Complex Analysis