The Cauchy-Riemann Conditions for Complex Variables

ECE 6382 y z z + x z z Fall 2023 David R. Jackson Notes 2 Differentiation of Functions of a Complex Variable Notes are adapted from D. R. Wilton, Dept. of ECE 1

Functions of a Complex Variable ( ) f z = Function of a complex variable: w = = x iy, w + = = + z u + iv ( ) f z ( ) ( ) = ( ( ) ) ( ) = + w u z iv z u x,y iv x,y ( ) f z ( ) 2 2 2 = = ( e.g., , ) 2 z u x,y x - y , v x,y xy Examples of functions: 2 = a bz + + = sinh( ) w cz , w A z + a bz dz + n = = w , w z 2 + c ez = 0 n 2

Differentiation of Functions of a Complex Variable Derivative of a function of a complex variable: ( ) z ( ) f z + f z z df dz f z ( ) z = = = lim z lim z f 0 0 To define a unique derivative at a point , the limit must exist at must be independent of the direction o z z ( ) = f at arg z z z ( ) + y f z z v z ( ) f z z = w z ( ) f z + x w z z f u 3

The Cauchy Riemann Conditions ( ) f z ( ) ( ) = + Denote z x i y = = + w u x,y iv x,y = First, let : z x ( ) x ( ) f z + f z x df dz f x = = lim x lim x x,y 0 u x 0 ( ) x ( ) ( ) x ( ) + + u x,y v x x,y v x,y = + lim x lim x i 0 0 Augustin-Louis Cauchy df dz u x v x = + i = Next let : z i y ( ) ( ) f z + f z i y i y df dz f = = lim y lim y + i y u x,y 0 0 ( ) y ( ) ( ) y ( ) + y u x,y v x,y y v x,y = + lim y lim y i i i 0 0 Bernhard Riemann df dz v y u y u x v x v y u y = + = Question: Is ? i i i 4

The Cauchy Riemann Conditions (cont.) We found y = z i y z df dz u x v x df dz v y u y z = + = i i = z x x = z i y = z x For a unique derivative, these expressions must be equal. That is, a for the existence of a derivative of function of a complex variable is that necessary condition u x v y u y v x = = & Cauchy-Riemann equations df dz We've proved that if exists Cauchy-Riemann conditions. 5

The Cauchy Riemann Conditions (cont.) df dz Next, we prove that Cauchy-Riemann conditions exists (sufficiency): u x u y v x v y + + + x y i x y + f z u i v z = + x i y u x v x u y v y + + + i x i y Arbitrary direction z = + v x x i y u x v x u x Use C.R. conditions + + + i x i y Totaldifferentials : = ( ) ( ) u x,y x v x,y x u x,y y v x,y y + x i y + ( ) u x,y x y u x v x u x v x + + + i y ( ) i x i ( ) ( ) + ( ) v x,y x y = + x i y + u x v x ( ) + x i y + i u x v x y x ( ) 1 = = + = , independent of arg tan i z x i y 6

The Cauchy Riemann Conditions (cont.) Hence, we have the following equivalent statements: df dz exists Cauchy-Riemann conditions. or df dz if and only if exists (iff) the Cauchy-Riemann conditions hold. or necessary and sufficient df dz The Cauchy-Riemann conditions are a condition for the e xistence of the derivative of a complex variable . f 7

The Cauchy Riemann Conditions (cont.) We say that a function is "analytic" at a point if the derivative exists there (and at all points in some neighborhood of the point). y ( ) f z D if the derivative exsits at each point in . is said to be "analytic" in a domain D D x The theory of complex variables largely exploits the remarkable properties of analytic functions. The terms " holomorphic", "regular", and "differentiable" are also used instead of "analytic." 8

Applying the Cauchy Riemann Conditions Example 1 : ( ) ( ) u x,y v x,y ( ) f z ( ) = = + = + z x iy x i y u x v y v x = = 1 ( ) C.R. conditions hold everywhere for finite z u y = = 0 is analytic everywhere z Example 2 : ( ) ( ) v x,y u x,y ( ) f z ( ) ( ) = = + = X + z* x iy * x i y u x v y = = 1 1 nowhere C.R. conditions hold u y z* v x = = is analytic nowhere 0 9

Applying the Cauchy Riemann Conditions (cont.) y Example 3 : 1 z 1 + x iy x + D ( ) f z = = = x 2 2 x iy y + x + y = + i 2 2 2 2 x y x y 1 z = is analytic except at 0 z ( ) ( ? ) u x,y v x,y : 0 D z 2 ( 2 2 + 2 ( 2 2 2 2 x y y + 2 ) u x x y x v y = = = ) 2 2 2 2 2 + + x y x y 2 u y xy v x = = = = = C.R. conditions hold everywhere except 0 ( 0) x y z . ( ) 2 2 2 + x y ( ) f z = = is analytic everywhere except at The point is called a "singularity." 0 0 z . z A singularity is a point where the function is not analytic. 10

Applying the Cauchy Riemann Conditions (cont.) Example 4 : ( ) f z ( ) z ( ) ( ) i iy ( ) iy ( ) iy = = + = + sin sin sin cos x sin cos x iy x ( ) i iy y y + + e e e e ( ) iy = = = bu t cos cosh y, 2 2 ( ) i iy ( ) i iy y y e e i iy e e ( ) ( ) z = = = sin sinh iy i i y 2 2 ( ) = + = + so sin sin sin cosh x sinh cos x y i y x ( ) ( ) u x,y v x,y C.R. conditions hold for all finite u x v y u y v x = = = = cos cosh x sin sinh x y , y z u x v x ( ) z = + = Now use: cos cosh x sin sinh x f i y i y ( ) ) iy ( ) iy = = = cos cos x sin sin x iy ( z + cos cos x d dz ( ) z ( ) z = = sin cos f z 11

Differentiation Rules (cont.) Example ( ) z ( ) ( ) z 2 2 2 ( ) ( ) z 2 2 + 2z z + z + z z ( ) z d dz 2 = = lim z lim z z z 0 0 = + 2 lim z z z 0 = 2 z Note:The above brute-force derivation, directly using the definition of the derivative, is exactly what is done in usual calculus, with x being used there instead of z. 12

Differentiation Rules It is relatively simple to prove on a case-by-case basis that practically all formulas for differentiating functions of real variables also apply to the corresponding function of a complex variable: n az sin dz cos dz d z dz d e dz d z d z 1 n az = = = = cos sin etc. nz , ae , z, z, ( ) z d dz ( ) z 1 n n = every polynomial of degree , in is analytic (differentiable). , nz N P N z ( ) ( ) P z Q z except every rational function ( ) Q z here in is analytic z w vanishes. 13

Differentiation Rules Replacing by in the usual derivations for functions of a real variable, we find practically all differentiation rules for functions of a complex variable turn out to be identical to t hose for real variables: x z ( ) ( ) ( ) d f z g z ( ) z ( ) g z = f dz ( ) ( ) ( ) dz f z dz g z d f z g z ( ) ( ) z g z ( ) f z g z ( ) = + f ( ) ( ) d g f f g = ( ) g 2 14

A Theorem Related to z* If f = f (z,z*) is analytic, then f = 0 z* (An analytic function cannot vary with z*, and therefore cannot be a function of z*, except in a trivial way.) All functions that contain z* are therefore not analytic, except for some trivial cases (where the function does not really vary with z*). 15

A Theorem Related to z* (cont.) Examples: f ( ) f z = = is analytic nowhere, since (not independent of ) * 1 0 * z z * z f z ( ) f z ( ) unless ( = = * 0 = + is not analytic, since ) sin * cos 2 1 / 2 z z z n * f ( ) f z 2 unless ( = = = = is not analytic, since ) * 0 0 z zz z z * z f ( ) z ( ) z ( ) f z ( ) z unless ( = + = = * * is not analytic, since ) sin cos sin 0 z n * z * * z z f ( ) f z ( ) 1 = = = = is analytic everywhere, since 1 0 * * z z 16

Proof of z* Theorem If f = f (z,z*) is analytic, then f z* C.R. conditions: u x v x v y u y = 0 = (An analytic function cannot vary with z*, and therefore cannot really be a function of z*, except in a trivial way.) = Note that = x iy + = x iy z , z* Treating and as independent variables: z z* C.R. conds. ( ) ( ) = u z,z* x v z,z* y u z z x u z* x = v z z y v z* y = = = = + = + z* z* 1 1 = i i z* u z u v z v + = (1) i z* 17

A Theorem Related to z* (cont.) Similarly, C.R. conds. = = ( ) ( ) v z,z* x u z,z* y v z z x v z* x = u z z y i u z* z* y = = + = z* = 1 1 = i v z v u z u z* + = (2) i z* Next, consider from (2) from (1) f u z* v v z v u z v z u z u z* = + = + + + + i i i z* z* z* u z* v f f = + = = 0 i z* z* z* f z * is independent of 18

Entire Functions A function that is analytic everywhere in the finite* complex plane is called entire . Typical functionsthat are entire (analytic everywhere in the finite complex plane): 2 3 4 5 n 1 , z, z , z , z ,z , ,z , z sin cos sinh cosh e , z, z, z, z almost Typical functions analytic everywhere: 1 z 1 1 2 / , tan cot tanh coth , z , z, z, z, z 2 2 z 1 * A function is said to be analytic everywhere in the finite complex plane if it is analytic everywhere except possibly at infinity. = Let 1 w / z Analytic at infinity: Is the function analytic at w = 0? 19

Combinations of Analytic Functions Combinations of functions: Finite linear combinations of analytic functions are analytic: are analytic is analytic ( ) f z ,g z ,h z af z ( ) ( ) ( ) If ( ) ( ) + + bg z ch z Composite combinations of analytic functions are analytic: ( ) f z ,g z ( ) ( f g z If are analytic ) ( ) is analy tic 20

Combinations of Analytic Functions (cont.) Infinite series: Infinite series may be: - Analytic everywhere - Analytic somwhere The somewhere might depend on the form used to represent the function. Example: 1 The first form is analytic everywhere except z = 1. ( ) f z = 1 z The second form is analytic for |z| < 1 (the series does not converge on or outside the unit circle). ( ) f z 2 3 = + + + + 1 1 z z z , z 21

Combination of Analytic Functions (cont.) Examples Composite functions of analytic functions are also analytic. ( ) ( ) g z = = 2 f z z sin z ( ) z ( ) ( ) ( ) = = 2 sin h z g f z analytic Derivatives of analytic functions are also analytic (proof given later). ( ) ( ) z = = sin f z z cos f z analytic 22

Derivatives of Analytic Function Important theorem (proven later) The derivative of an analytic function is also analytic. ( ) f z is analytic ( ) z f is analytic Hence, all derivatives of an analytic function are also analytic. ( ) z f is analytic 23

Real and Imaginary Parts of Analytic Functions Are Harmonic Functions ( ) f z ( ) ( ) = + Assume an analytic function: u x,y iv x,y ( ) ( ) 2 2 = = 0 u x,y v x,y The functions u and v are harmonic (i.e., they satisfy Laplace s equation) ( ) ( ) v z ( ) = = = = u u z u x,y Notation: ( ) v v x,y This result is extensively used in conformal mapping to solve electrostatics and other problems involving the 2D Laplace equation (discussed later). Pierre-Simon Laplace 24

Real and Imaginary Parts of Analytic Functions Are Harmonic Functions (cont.) Proof f is analytic df / dzis also analytic (see slide 23) df dz u x f v x v y u y ( ) ( ) ( ) ( ) = + = = + Notation Analytic : f z u x,y iv x,y i i ( ) z + Denote U iV u x v y v x u y ( ) ( ) ( ) z ( ) = = = = = We have: Re ; U x,y f i V x,y i i ( ) z Apply the C.R. conditions to : f 2 2 U x V y u u 2 = 0 u = = 2 2 x y 2 2 V x U y v v 2 = 0 v = = 2 2 x y 25

Real and Imaginary Parts of Analytic Functions Are Harmonic Functions (cont.) ( ) f z 2 = = Example: w z ( ) ( ) ( ) 2 2 2 = + = + = + 2 w u iv x iy x y i xy ( ( ) ) 2 2 = u x,y x y = 2 v x,y xy 2 2 2 2 u u v v 2 2 = + = = = + = + = 2 2 0 u 0 0 0 v 2 2 2 2 x y x y 26

Real and Imaginary Parts of Analytic Functions Are Harmonic Functions (cont.) ( ) f z ( ) z = = sin w Example: ( ) = + = x iy + = y i + sin sin cosh x cos sinh x w u iv y ( ( ) ) = , sin cosh x u x y y = , cos sinh x v x y y 2 2 u u = + = + = 2 sin cosh x sin cosh x 0 u y y 2 2 x y 2 2 v v = + = + = 2 cos sinh x cos sinh x 0 v y y 2 2 x y 27