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The Cauchy Riemann (CR) conditions provide a necessary and sufficient condition for a function f(z) = u(x, y) + iv(x, y) to be analytic in a region. The CR conditions require that the partial derivatives of u and v satisfy u/x = v/y and u/y = -v/x. If a function satisfies these conditions at all points in a region, then it is analytic in that region. The document proves this using cases where y = 0 and x = 0, showing the derivatives must be equal. Examples are provided to demonstrate checking functions for analytic
![NPTEL Physics Mathematical Physics - 1
Lecture 33
Cauchy Riemann (CR) conditions
CR conditions state that a necessary and a sufficient condition that a
function,
() defined as,
() = (, ) + (, )
be analytic in a region provides the following conditions (called CR conditions) on the
partial derivatives of are met,
=
and
=
in . Additionally it is assumed that the partial derivatives are continuous in .
To arrive at the proof of the above equations, let us look at the necessary conditions. For
a function () to be analytic, it should be differentiable in a region , that is
瑞
0
(+()()
= ()
( + , + ) + ( + , + ) (, ) + (, )
+
モ0
瑞
0
瑞
must exist and should be independent of the manner in approaches zero. Two
possibilities are apparent for that to happen.
Case A
= 0, 0
For which () is defined as,
() = モ0
瑞 ( + , ) (, )
+ [
( + , )(, )
]
= +
Assuming that the partial derivatives exist.
Joint initiative of IITs and IISc Funded by MHRD Page 12 of 66](https://image.slidesharecdn.com/lec33-231023101222-f1d15af4/85/lec33-ppt-1-320.jpg)
![NPTEL Physics Mathematical Physics - 1
Case B
0, = 0
In this case,
() = 0
瑞 (, + ) (, )
+
(, + ) (, )
=
+
Of course, these two cases will have to yield identical results. Thus
+
=
+
Or, =
and =
Next we look at the sufficiency conditions. Since and
are assumed to be
continuous, we have
= ( + , + ) (, )
= [( + , + ) (, + )] + [(, + ) (, )]
Where the second and third terms are subtracted and added.
Hence =
+
Similarly =
+
Thus = + = ( + ) + (
+ )
Changing the second bracket using CR conditions
= ( + ) + ( + )
= ( + ) ( + )
Joint initiative of IITs and IISc Funded by MHRD Page 13 of 66](https://image.slidesharecdn.com/lec33-231023101222-f1d15af4/85/lec33-ppt-2-320.jpg)
lec33.ppt
- 1. NPTEL Physics Mathematical Physics - 1 Lecture 33 Cauchy Riemann (CR) conditions CR conditions state that a necessary and a sufficient condition that a function, () defined as, () = (, ) + (, ) be analytic in a region provides the following conditions (called CR conditions) on the partial derivatives of are met, = and = in . Additionally it is assumed that the partial derivatives are continuous in . To arrive at the proof of the above equations, let us look at the necessary conditions. For a function () to be analytic, it should be differentiable in a region , that is 瑞 0 (+()() = () ( + , + ) + ( + , + ) (, ) + (, ) + モ0 瑞 0 瑞 must exist and should be independent of the manner in approaches zero. Two possibilities are apparent for that to happen. Case A = 0, 0 For which () is defined as, () = モ0 瑞 ( + , ) (, ) + [ ( + , )(, ) ] = + Assuming that the partial derivatives exist. Joint initiative of IITs and IISc Funded by MHRD Page 12 of 66
- 2. NPTEL Physics Mathematical Physics - 1 Case B 0, = 0 In this case, () = 0 瑞 (, + ) (, ) + (, + ) (, ) = + Of course, these two cases will have to yield identical results. Thus + = + Or, = and = Next we look at the sufficiency conditions. Since and are assumed to be continuous, we have = ( + , + ) (, ) = [( + , + ) (, + )] + [(, + ) (, )] Where the second and third terms are subtracted and added. Hence = + Similarly = + Thus = + = ( + ) + ( + ) Changing the second bracket using CR conditions = ( + ) + ( + ) = ( + ) ( + ) Joint initiative of IITs and IISc Funded by MHRD Page 13 of 66
- 3. NPTEL Physics Mathematical Physics - 1 = ( + ) Hence 瑞 р0 = = () = + proving that the derivative exists and so () is analytic . Examples 1. () = 2 = 2 2 + 2ヰ () = 2 To verify CR condition, (, ) = 2 2 (, ) = 2ヰ = 2 = ｐ , = 2 = ｐ () = 2 + 2 = 2 (where the suffix refers to the derivative taken with respect to that variable). 2. () = ||2, (, ) = 2 + 2, (, ) = 0 If the CR conditions hold at a point (, ) then the point would be, 2 = 0, 2 = 0 (, ) = (0,0). So () does not exist at a non-zero point. Exercise: Check for the analyticity of the following functions. () = ||3 || = 2 + 2 ||3 = (2 + 2)2 3 3 () = (2 + 2)2 3 (, ) = 0, (, ) = (2 + 2)2 It is a purely complex function. Its differentiable only at the origin and hence not an analytic function. Joint initiative of IITs and IISc Funded by MHRD Page 14 of 66
- 4. Joint initiative of IITs and IISc Funded by MHRD Page 15 of 66 NPTEL Physics Mathematical Physics - 1 Second Theorem Sufficient condition for differentiability Let the function () = (, ) + (, ) be defined throughout some neighborhood of a point 0 = 0 + 0 and suppose that the first-order partial derivatives of the functions and with respect to and exist everywhere in that neighborhood. If those partial derivatives are continuous at (0, 0) and satisfy CR conditions, then = ｐ, = ｐ at (0, 0) then (0, 0) (= f (0)) exists. = ｐ, ｐ = should necessarily be satisfied if () has a derivative at a point z. Thus it is the necessary and sufficient condition that must hold if () is differentiable. If they hold, then () exists and is given by, () = (, ) + ｐ(, ) () = (, ) + ｐ(, )