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1 Arc of a circle is part of its circumference

2 Both circles are concentric if they have the same centre

3 Circumference of a circle is 2*pi*radius or pi*dianeter

4 Diameter cuts through a circle's centre and is its longest chord

5 Exact value of pi is not known

6 Four right angles can be found in a circle

7 Gross surface area of a circle is pi*radius2

8 Half a circle is a semi-circle

9 Just an estimate is usually used for the value of pi

10 Knowledge of a circle's properties and features were known by the ancients

11 Inner circumference is less than its outer circumference

12 Like dimensional circles are congruent

13 Motors cars and machinery depend on a circle to move and operate

14 No computer has ever worked out a circle's circumference/diameter

15 One rotation of a circle goes through 360 degrees

16 Perpendicular lines are formed when its diameters meet at right angles

17 Quadrilaterals and their 4 sides have cyclic properties within a circle

18 Radius is 1/2 of a circle's diameter

19 Sectors and segments are found within a circle

20 Tangent is an outside straight line that touches a circle at 1 point

21 Unlimited lines of symmetry

22 Vertex of a circle doesn't exist

23 Wheels are a circle's best friend because they have so much in common

24 X as a letter fits perfectly into a circle creating 4 sectors

25 Yo-yo is a circular device that rolls up and down on a string

26 Zero coordinates of (0, 0) can be the centre of a circle on the Cartesian plane

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Since z = a^2/z on the circle, we see that w as given by (1) is purely real on the circle C and therefore if* = 0. Thus C is a streamline. If the point z is outside 0, the point az /z is inside 0, and vice-versa. Since all the singularities off(z) are by hypothesis exterior to C, all the singularities off(a?/z) are interior to C ; in particular f(a z /z) has no singularity at infinity, since f(z) has none at z = 0. Thus w has exactly the same singularities as/(z) and so all the conditions are satisfied.


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To apply the transformation ( w = z + \frac{1}{z} ) to the circle defined by ( |z| = 2 ), we can express ( z ) in polar form as ( z = 2e^{i\theta} ), where ( \theta ) ranges from ( 0 ) to ( 2\pi ). Substituting this into the equation for ( w ), we get ( w = 2e^{i\theta} + \frac{1}{2e^{i\theta}} = 2e^{i\theta} + \frac{1}{2} e^{-i\theta} ). This simplifies to ( w = 2e^{i\theta} + \frac{1}{2}(\cos \theta - i \sin \theta) ), which describes a new curve in the ( w )-plane. The resulting curve can be analyzed further to understand its geometric properties.


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