Example 3.10.  Show that, if  [Graphics:Images/CauchyRiemannMod_gr_1203.gif]  is is the principal square root function given by  

                    [Graphics:Images/CauchyRiemannMod_gr_1204.gif],     

where the domain is restricted to be   [Graphics:Images/CauchyRiemannMod_gr_1205.gif],   then the derivative is given by  

                    [Graphics:Images/CauchyRiemannMod_gr_1206.gif],    

for every point in the domain   [Graphics:Images/CauchyRiemannMod_gr_1207.gif].  

 

Explore Solution 3.10.

 

Solution.  We write  

                    [Graphics:../Images/CauchyRiemannMod_gr_1220.gif],     

                    and   
            
                    [Graphics:../Images/CauchyRiemannMod_gr_1221.gif].  
                    
Thus,

                    [Graphics:../Images/CauchyRiemannMod_gr_1222.gif],     

                    and   

                    [Graphics:../Images/CauchyRiemannMod_gr_1223.gif].  

Moreover, the partial derivatives  [Graphics:../Images/CauchyRiemannMod_gr_1224.gif]   are continuous in the domain   

[Graphics:../Images/CauchyRiemannMod_gr_1225.gif]    (note the strict inequality in  [Graphics:../Images/CauchyRiemannMod_gr_1226.gif]).   

By Theorem 3.5,   [Graphics:../Images/CauchyRiemannMod_gr_1227.gif],   is differentiable in the domain   

[Graphics:../Images/CauchyRiemannMod_gr_1228.gif].   Therefore, using Equation (3-23) and (3-24), we have    

                    [Graphics:../Images/CauchyRiemannMod_gr_1229.gif]  

And an alternative calculation is  

                    [Graphics:../Images/CauchyRiemannMod_gr_1230.gif]  

Note that   [Graphics:../Images/CauchyRiemannMod_gr_1231.gif]   is discontinuous on the negative real axis and is undefined at the origin.  

Using the terminology of Section 2.4, the negative real axis is a branch cut, and the origin is a branch point for this function.  

 

We are done.

 

Aside.  Both [Graphics:../Images/CauchyRiemannMod_gr_1232.gif] and [Graphics:../Images/CauchyRiemannMod_gr_1233.gif] can assist us in finding the partial derivatives.  

Aside.  The Mathematica solution uses the commands.  

[Graphics:../Images/CauchyRiemannMod_gr_1234.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1235.gif]


[Graphics:../Images/CauchyRiemannMod_gr_1236.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1237.gif]


[Graphics:../Images/CauchyRiemannMod_gr_1238.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1239.gif]


[Graphics:../Images/CauchyRiemannMod_gr_1240.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1241.gif]


[Graphics:../Images/CauchyRiemannMod_gr_1242.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1243.gif]


[Graphics:../Images/CauchyRiemannMod_gr_1244.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1245.gif]


[Graphics:../Images/CauchyRiemannMod_gr_1246.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1247.gif]


[Graphics:../Images/CauchyRiemannMod_gr_1248.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1249.gif]

Mathematica shows that the polar form of the Cauchy-Riemann equations (3-22) are satisfied for all   [Graphics:../Images/CauchyRiemannMod_gr_1250.gif],    

                    [Graphics:../Images/CauchyRiemannMod_gr_1251.gif],     

                    and     
                    
                    [Graphics:../Images/CauchyRiemannMod_gr_1252.gif].  

Aside.  The Maple commands are similar.  

          >  [Graphics:../Images/CauchyRiemannMod_gr_1253.gif]

                              [Graphics:../Images/CauchyRiemannMod_gr_1254.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_1255.gif]

                              [Graphics:../Images/CauchyRiemannMod_gr_1256.gif]


          >  [Graphics:../Images/CauchyRiemannMod_gr_1257.gif]

                              [Graphics:../Images/CauchyRiemannMod_gr_1258.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_1259.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_1260.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_1261.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_1262.gif]


          >  [Graphics:../Images/CauchyRiemannMod_gr_1263.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_1264.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_1265.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_1266.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_1267.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_1268.gif]

 

Maple shows that the polar form of the Cauchy-Riemann equations (3-22) are satisfied for all   [Graphics:../Images/CauchyRiemannMod_gr_1269.gif],    

                    [Graphics:../Images/CauchyRiemannMod_gr_1270.gif],     

                    and     
                    
                    [Graphics:../Images/CauchyRiemannMod_gr_1271.gif].  

The Cauchy-Riemann equations hold  all points  [Graphics:../Images/CauchyRiemannMod_gr_1272.gif]  in the complex plane,

therefore   [Graphics:../Images/CauchyRiemannMod_gr_1273.gif]   is an analytic function,

for all  except at points that lie on the negative [Graphics:../Images/CauchyRiemannMod_gr_1274.gif]-axis and   [Graphics:../Images/CauchyRiemannMod_gr_1275.gif].  

Verify that the derivative can be calculated with either of the formulas:

(3-23)              [Graphics:../Images/CauchyRiemannMod_gr_1276.gif],     

                        or    

(3-24)              [Graphics:../Images/CauchyRiemannMod_gr_1277.gif].  

Aside.  The Mathematica solution uses the commands.  

[Graphics:../Images/CauchyRiemannMod_gr_1278.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1279.gif]


[Graphics:../Images/CauchyRiemannMod_gr_1280.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1281.gif]


[Graphics:../Images/CauchyRiemannMod_gr_1282.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1283.gif]


[Graphics:../Images/CauchyRiemannMod_gr_1284.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1285.gif]


[Graphics:../Images/CauchyRiemannMod_gr_1286.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1287.gif]

Aside.  The Maple commands are similar.  

          >  [Graphics:../Images/CauchyRiemannMod_gr_1288.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_1289.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_1290.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_1291.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_1292.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_1293.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_1294.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_1295.gif]

          >  [Graphics:../Images/CauchyRiemannMod_gr_1296.gif]  

                              [Graphics:../Images/CauchyRiemannMod_gr_1297.gif]

Both Mathematica and Maple have shown that if    

                    [Graphics:../Images/CauchyRiemannMod_gr_1298.gif] ,  

where the domain is restricted to be   [Graphics:../Images/CauchyRiemannMod_gr_1299.gif],   then the derivative is given by   

                    [Graphics:../Images/CauchyRiemannMod_gr_1300.gif],  

where   [Graphics:../Images/CauchyRiemannMod_gr_1301.gif].  

 

We are really done.

 

Aside.  Figure E.3.10 a, shows the graphs of   [Graphics:../Images/CauchyRiemannMod_gr_1302.gif]   and   [Graphics:../Images/CauchyRiemannMod_gr_1303.gif].  

The partial derivatives of  [Graphics:../Images/CauchyRiemannMod_gr_1304.gif]  are  

                    [Graphics:../Images/CauchyRiemannMod_gr_1305.gif]     and     [Graphics:../Images/CauchyRiemannMod_gr_1306.gif],   

and the partial derivatives of  [Graphics:../Images/CauchyRiemannMod_gr_1307.gif]  are

                    [Graphics:../Images/CauchyRiemannMod_gr_1308.gif]     and     [Graphics:../Images/CauchyRiemannMod_gr_1309.gif].  

        They satisfy the Cauchy-Riemann equations (3-22) because they are the real and imaginary parts of an analytic function.  
        
At the point   [Graphics:../Images/CauchyRiemannMod_gr_1310.gif],   we have   [Graphics:../Images/CauchyRiemannMod_gr_1311.gif]   and   [Graphics:../Images/CauchyRiemannMod_gr_1312.gif],   and these partial derivatives

appear along the edges of the surfaces for  [Graphics:../Images/CauchyRiemannMod_gr_1313.gif]  at the points   [Graphics:../Images/CauchyRiemannMod_gr_1314.gif]   and   [Graphics:../Images/CauchyRiemannMod_gr_1315.gif],   

respectively.   Similarly,  at the point  [Graphics:../Images/CauchyRiemannMod_gr_1316.gif],   we have   [Graphics:../Images/CauchyRiemannMod_gr_1317.gif]   and   [Graphics:../Images/CauchyRiemannMod_gr_1318.gif]   and these

partial derivatives appear along the edges of the surfaces for  [Graphics:../Images/CauchyRiemannMod_gr_1319.gif]  at the points   [Graphics:../Images/CauchyRiemannMod_gr_1320.gif]   and  

[Graphics:../Images/CauchyRiemannMod_gr_1321.gif],   respectively.

 

[Graphics:../Images/CauchyRiemannMod_gr_1322.gif]     [Graphics:../Images/CauchyRiemannMod_gr_1323.gif]  

                  [Graphics:../Images/CauchyRiemannMod_gr_1324.gif].                                                          [Graphics:../Images/CauchyRiemannMod_gr_1325.gif].  
                                                                        Figure E.3.10 a

 

[Graphics:../Images/CauchyRiemannMod_gr_1326.gif]     [Graphics:../Images/CauchyRiemannMod_gr_1327.gif]  

                  [Graphics:../Images/CauchyRiemannMod_gr_1328.gif],   and                                               [Graphics:../Images/CauchyRiemannMod_gr_1329.gif],   and  

                  [Graphics:../Images/CauchyRiemannMod_gr_1330.gif].                                                             [Graphics:../Images/CauchyRiemannMod_gr_1331.gif].  
                                                                        Figure E.3.10 b

 

[Graphics:../Images/CauchyRiemannMod_gr_1332.gif]     [Graphics:../Images/CauchyRiemannMod_gr_1333.gif]  

                  [Graphics:../Images/CauchyRiemannMod_gr_1334.gif],   and                                             [Graphics:../Images/CauchyRiemannMod_gr_1335.gif],   and  

                  [Graphics:../Images/CauchyRiemannMod_gr_1336.gif].                                                          [Graphics:../Images/CauchyRiemannMod_gr_1337.gif].  
                            
                  Remark. It is difficult to visualize  [Graphics:../Images/CauchyRiemannMod_gr_1338.gif]  and  [Graphics:../Images/CauchyRiemannMod_gr_1339.gif]  because these partial derivatives  

                  are taken with respect to changes in the polar angle  [Graphics:../Images/CauchyRiemannMod_gr_1340.gif],  and so they cannot be visualized as an "ordinary slope."


                                                                        Figure E.3.10 c

 

                                        For the function   [Graphics:../Images/CauchyRiemannMod_gr_1341.gif]   we see that

                                        [Graphics:../Images/CauchyRiemannMod_gr_1342.gif],     and     
                                                  
                                        [Graphics:../Images/CauchyRiemannMod_gr_1343.gif].  

                                                                        Figure E.3.10

 

We are really really done.

 

Aside.  We can let Mathematica check out the calculations given above.  

[Graphics:../Images/CauchyRiemannMod_gr_1344.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1345.gif]


[Graphics:../Images/CauchyRiemannMod_gr_1346.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1347.gif]


[Graphics:../Images/CauchyRiemannMod_gr_1348.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1349.gif]


[Graphics:../Images/CauchyRiemannMod_gr_1350.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1351.gif]


[Graphics:../Images/CauchyRiemannMod_gr_1352.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1353.gif]


[Graphics:../Images/CauchyRiemannMod_gr_1354.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1355.gif]


[Graphics:../Images/CauchyRiemannMod_gr_1356.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1357.gif]


[Graphics:../Images/CauchyRiemannMod_gr_1358.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1359.gif]


[Graphics:../Images/CauchyRiemannMod_gr_1360.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1361.gif]


[Graphics:../Images/CauchyRiemannMod_gr_1362.gif]

[Graphics:../Images/CauchyRiemannMod_gr_1363.gif]

Remark.  In this book the use of computers is optional.  

Hopefully this text will promote their use and understanding.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

This solution is complements of the authors.

 

 

 

 

 

 

 

 

 

 

 

This material is coordinated with our book Complex Analysis for Mathematics and Engineering.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

(c) 2011 John H. Mathews, Russell W. Howell