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sin x cosh y i cos x sinh y Similarly, cos z cos x iy ei x iy e i x iy 2 1 feix y e ix y g 1 fe y cos x i sin x ey cos x i sin x g 2 2  y   y  e e y e e y cos x i sin x cos x cosh y i sin x sinh y 2 2
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DERIVATIVES, CAUCHY-RIEMANN EQUATIONS 16.5. Prove that d " " z, where z is the conjugate of z, does not exist anywhere. dz
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d f z z f z f z lim if this limit exists independent of the manner in which z!0 dz z z x i y approaches zero. Then By de nition,
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CHAP. 16]
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FUNCTIONS OF A COMPLEX VARIABLE
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" d z z z x iy x i y x iy " lim z lim x!0 z!0 dz z x i y y!0 lim
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x!0 y!0
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x iy x i y x iy x i y lim x!0 x i y x i y y!0 lim x 1: x i y 1: i y
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If y 0, the required limit is If x 0, the required limit is
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x!0
y!0
These two possible approaches show that the limit depends on the manner in which z ! 0, so that the " derivative does not exist; i.e., z is nonanalytic anywhere.
16.6. (a) If w f z
1 z dw , nd . (b) Determine where w is nonanalytic. 1 z dz
Method 1:
1 z z 1 z dw 2 1 z z 1 z lim lim z!0 1 z z 1 z dz z!0 z 2 provided z 6 1, independent of the manner in which z ! 0: 1 z 2 Thus, by the quotient rule for
Method 2. The usual rules of di erentiation apply provided z 6 1. di erentiation,
  1 z d 1 z 1 z d 1 z d 1 z 1 z 1 1 z 1 2 dz dz dz 1 z 1 z 2 1 z 2 1 z 2 (b) The function is analytic everywhere except at z 1, where the derivative does not exist; i.e., the function is nonanalytic at z 1.
16.7. Prove that a necessary condition for w f z u x; y i v x; y to be analytic in a region is that @u @v @u @v , be satis ed in the region. the Cauchy-Riemann equations @x @y @y @x
Since f z f x iy u x; y i v x; y , we have f z z f x x i y y u x x; y y i v x x; y y Then
z!0
f z z f z u x x; y y u x; y ifv x x; y y v x; y g lim x!0 z x i y y!0 & ' u x x; y u x; y v x x; y v x; y @u @v i i x x @x @x & ' u x; y y u x; y v x; y y v x; y 1 @u @v i y y i @y @y @u @v 1 @u @v @u @v i i @x @x i @y @y @y @y
If y 0, the required limit is
x!0
If x 0, the required limit is
y!0
If the derivative is to exist, these two special limits must be equal, i.e.,
FUNCTIONS OF A COMPLEX VARIABLE @u @v @v @u and : @x @y @x @y
[CHAP. 16
so that we must have
Conversely, we can prove that if the rst partial derivatives of u and v with respect to x and y are continuous in a region, then the Cauchy Riemann equations provide su cient conditions for f z to be analytic.
16.8. (a) If f z u x; y i v x; y is analytic in a region r, prove that the one parameter families of curves u x; y C1 and v x; y C2 are orthogonal families. (b) Illustrate by using f z z2 .
(a) Consider any two particular members of these families u x; y u0 ; v x; y v0 which intersect at the point x0 ; y0 . Since du ux dx uy dy 0, we have dy u x: dx uy
Also since dv vx dx vy dy 0;
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