Lecture 4 Properties of Logarithmic Function (Contd ) Since, Logln iarg u Re Log ln( ) v Im Log tan constant It follows that u v, u v This shows that Re Logand Im Log are (i) continuous in C { :Re 0,Im 0} (ii) partiall differentiable and first order partial derivatives are continuous in C { :Re 0,Im 0} (iii) Cauch Riemann equations hold. Therefore, Log is analtic in C { :Re 0,Im 0} and d i Log u i v. d The branch of logarithm log, with 0 0 arg 0, is a single valued function, and its properties are similar to the above properties of Log.
Eponential Function. Define e e (cos isin ) Note that e and e 0 as. But, lim e i does not eist, since cosn takes the values and for even and odd n, respectivel. Consequentl, lim e does not eist. It follows easil b using the truth of CR equations for and the continuit of first order partial derivatives of real and imaginar parts of e, that e is analtic for all and d e e. d Since, e e 0, it follows that e 0 for an. e e is a periodic function of comple period i, since i i( ) e e e.
The diagram sketched below illustrates that the fundamental period strip { i :, } is mapped oneone on to C {0} b the function e. 3 e 0 0 w The above diagram is eplained b the following analtical arguments: we (cos isin ) u e cos, v e sin, e u v or ln w, w0 and v v v ArgwTan ; Tan or Tan u u u ( According as u 0; u 0, v 0or u 0, v 0) a unique (, ), with,, such that Log w e w.
4 Log is inverse function of Log e. e, since Log e and In fact, that an branch of logarithm is inverse of eponential function, can be seen as follows: Let, r(cos isin ) i. Then, log ln i arg, where arg, with 0 0. 0 log ln i(arg ) e 0 e. e Similarl, 0 ln e. i e. i e log e ln e i arg e, where, k0 arg ( e ) k0 for some integer k 0. ln e i =.
5 Note: After introducing Power Series in the sequel we will be able to prove that (i)if f is differentiable in the entire comple plane C, f(0) = and f ( ) f( ) for all, then f() is the eponential function. (ii) If f is differentiable in the entire comple plane C, f(0) = f (0) = and f ( ) f( ). f( ) for all and, then f() is an eponential function. Another characteriation of the eponential function can be found in G. P. Kapoor, A new characteriation of the eponential function, Amer. Math. Monthl (974).
6 Trigonometric and Hperbolic Functions. The definitions of Trigonometric and Hperbolic Functions of comple valued functions of a comple variable as given below are analogous to corresponding Trigonometric and Hperbolic Functions of real valued functions of a real variable. However, some of properties of Trigonometric and Hperbolic functions of a comple variable as pointed out below are drasticall different from corresponding functions of a real variable. Definitions i i i i e e e e cos, sin, i cosec sin, sec cos. tan sin, cos cot cos, sin e e e e cosh cosi, sinh isini, sinh cosh tanh itani, coth icot i, cosh sinh cosech sinh, sech cosh
7 Properties: The following properties of Trigonometric and Hperbolic functions of a comple variable are drasticall different from corresponding functions of a real variable, rest of standard properties are analogous:. sin sin cosh icos sinh, cos cos cosh isin sinh, for i. Similar identities for other Trigonometric and Hperbolic functions can also be easil derived.. sin sin sinh, i cos cos sinh for 3. sin andcosare unbounded functions in C (A comple valued function f( ) is said to be bounded in a set A if f ( ) M for some M and all A, otherwise it is said to be unbounded). Since, sini and cosi as, sin andcosare unbounded functions in C. Recall that, since sin and cos for all R, sin andcos are bounded functions in R.
8 Harmonic Conjugate: Let u : CR be a harmonic function, i.e. the function u and its partial derivatives up to the second order are continuous and satisf the Laplace s equation u u 0. Definition: A function v : CR is said to be Harmonic Conjugate of harmonic function u : CR, if u v and v u. The following Leibnit Rule of differentiation under integral sign is needed for proving the eistence of harmonic conjugate: Let :[ ab, ] [ cd, ] C be continuous. Define If t b g() t (,) s t ds. a eists and is cont. on [ ab, ] [ cd, ], then g is diff. & g() t b a (,) s t ds t
Theorem (Eistence of Harmonic Conjugates). 9 Let G B(0, R), 0 R and u: GR be harmonic. Then, u has a harmonic conjugate in G Proof. Define v(, ) b v (, ) u( t, ) dt( ) 0 and determine ( ) such that v u. 0 (, ) L L R (Note that v u is obviousl satisfied) The above integral is well defined since L B(0, R). Using Leibnit Rule, v (, ) u (, t) dt( ) 0 0 u (, t) dt( ) = u(, ) u(,0) ( ) ( ) u (,0) (sin ce v u ) Consequentl, v(, ) u (, t) dt u ( s,0) ds 0 0 (since L B(0, R) ) is the required harmonic conjugate.
0 Notes.. The proof of above theorem gives a method to construct harmonic conjugate of a given harmonic function in the disk B(0, R), 0 R. The arguments of the proof and the result of the above theorem are valid for an domain G that is conve both in the direction of and direction of. 3. If G B(, c R), where c = (a, b), the above arguments could be modified to give the harmonic conjugate as: b a v (, ) u( t, ) dt u( sb, ) ds
Remarks.. Harmonic conjugate of u is unique up to a constant (To see this, let v and w be two harmonic conjugates of u. Then, u v w u v w Therefore, v w v w( ) v w v w ( ) so that ( ) ( ) constt.. The function f, whose real part is u is determined uniquel up to purel imaginar constant (Since harmonic conjugate v of u is unique up to a real constant, f = u + i v is uniquel determined up to a purel imaginar constant.)
Eamples.. u (, ). Since, u (, ), u (, ), the harmonic conjugate is v (, ) u( t, ) dt u( s,0) ds 0 0. dt 0 ds 0 0. u (, ). Since u (, ), u (, ), the harmonic conjugate is v (, ) u( t, ) dt u( s,0) ds 0 0 tdt sds. 0 0 The corresponding analtic function is f ( ) i( ) i(( ) i) i
3 Other Methods to find Harmonic Conjugates Method. If u is harmonic in a region contained in {: > 0} (i.e., > 0, > 0 or first quadrant) and homogenous of degree m, m 0, i.e. for an t > 0, ( ) m ut t u ( ), then v ( u u ) m is a conjugate harmonic function of u. Proof. Since u (, ) is a homogenous function, b Euler s Formula (see the derivation after this proof), u (, ) ( u u ) m To show that v ( u u ) is the harmonic conjugate. m It is easil verified that u ( u u u ) m v ( u u u ) m u v (since, u is continuous and u satisfies Laplace s equation. The equation u v is verified similarl.
4 Eample. Find an analtic function whose real part is u (, ). The function u is homogenous of degree and harmonic for all = + i. Therefore, b Method above, v (, ) [ ( ) ( )] ( ) The corresponding analtic function is therefore f ( ) uiv ( i).
5 Derivation of Euler s Formula for Homogenous Functins: d ut mt m u ( ( )) ( ( )) dt where t, t. u u t t mt m u( ), Now, make t and use continuit of u, to get u ( h) u ( ) u ( h) u ( ) u u and u u( ) h h as and
6 Method 3 (Milne Thompson Method: A completel informal method). Let u (, ) be a given Harmonic function. In the given epression of u, (, ) put, (!) and i consider g ( ) u(, ) f(0). i The imaginar part of g ( ) is the desired harmonic conjugate of u. (, ) Eample. u (, ) Using Milne Thompson method, f( ) u(, ) u(0,0) i [( ) ( ) ] i Thus the desired harmonic conjugate is v (, ) Im f( ).
Informal Justification of Milne Thompson Method: 7 Let v (, ) be a Harmonic conjugate of the given Harmonic function u (, ) and g u iv be the corresponding analtic function, Denote, ( i ). Then, g( ) ( u iv) ( i )( u iv) [ u iv i ( u iv )] [ u iv iu v ] [ u v i ( v u )] 0 () Informall assuming that, are independent variables (!), deduce from () that f ( ) is independent of, i.e. it is a function of alone, i.e. * g ( ) g( ) ( sa). u (, ) [ g ( ) g ( )] [ ( ) * ( )] () gg
8 In (), = + i, where, and are real. Let us informall assume that () holds as well with and comple (!) and put,. i Then, () gives * u(, ) [ g( ) g (0)] ( since i i 0) i i [ ( ) (0)] (3) g g Equation (3) g( ) u(, ) g(0) i u(, ) u(0,0) u(0,0) g(0) i Since u(0,0) f(0) is a purel imaginar constant, it can be dropped from the above epression (since harmonic conjugates are unique onl up to an imaginar constant). g ( ) u(, ) f(0) i is an analtic function whose real part is u. (, ) The imaginar part of g ( ) is therefore the desired harmonic conjugate of u. (, )