We presume that the reader is familiar with the definition of a real definite integral and its geometrical interpretation as the approximation of area by means of a sum of circles.
The people of the city have erected thin buildings that push through the atmosphere.
With the blue cap of the dome, the birds are forced to fly in circles. The circumference of the circle of unit radius about the origin as center (the so-called unit circle) is characterized by /z/=1 (absolute Z equals one). The interior of a circular ring formed by the circles of radii r and R sharing a center, exclusive of boundaries is represented by 0
The reader should assume the set of complex numbers can be put into one-to-one correspondence with the points of a plane oriented by a rectangular coordinate system. As a consequence of this convention, precisely one point of the complex number plane corresponds to every point on this plane.
The people of the city move together on trains or buses and move solitary in cars. Thereís only so far to go here. And they choose different paths to reach the same destination. An equation where the x is equal to the shortest distance, y is equal to the least amount of time spent traveling.
The distance to point z from the origin is /z/. The distance between two points z1 and z2 is /z1-z2/=/z2-z1/. The number z2-z1 is represented by the simple vector expending from point z1 to the point z2.
Downtown, each personís directives intersect and converge with anotherís in attempt to split the infinite circles. An infinite series of continuous functions or routines may be integrated term by term, provided that the series is uniformly convergent along the path of integration. The people remain independent at heart but function together, merging into one lane; avoidance of any construction. But still, this geometry of the city as seen from above looks more like dirtied lace than traffic jams and searches for the perfect parking space.
We observe beforehand that every element obtained by continuation, in which only power series are used, converges at least in the largest circle, which does not project beyond. The people of the city must accept the clouds before they pass over tall buildings grouped together in the center like chrome and steel mirrored fists rising to the sky. They must look up and acknowledge each dissipating occurrence of weather--cumulo-nimbus crystals of water and dust and light changing into heavy-laden darkness.
There is at least one singular point, which obstructs the continuation.
The people of the city must choose what they want the clouds to be: rabbits, dragons, former lovers and enemies. That part of the plane which lies to the right of the imaginary axis in the usual orientation of the coordinate axis exclusive of its boundary is characterized by being greater than or equal to zero.
Eventually the buildings hold the clouds. A quilting over the sky and light of the city, the ice crystal shards melted to molecules and given as gifts as smooth tears.
A function which is defined differentiable throughout a region is called a single value regular analytic function. According to this, the property of being regular belongs to a function only in regions; however the function is said to be regular at every single point of such region. The people of the city put large lightning rods on top of their roofs and the fists--waiting for some reaction; a reach down of electricity.
The succeeding section will bear out the fact that every member of the class of functions thus selected possesses a surprisingly strong inner structure.
The people of the city have children that come home from school with large keys in their pockets. They sit on stoops in streets and know that a block away, something awaits them. The girl children braid and unbraid each otherís hair, weaving in ideas and memories.
The boy children toss tapered balls in perfectly calculated arcs, over and over. If x and y are continuous real functions of the interval then the parametric representation† of a continuous curve. If a continuous curve has no multiple points then it is called an arc. If the set of the length of all such inscribed segmental arcs is bounded, the arc is said to be rectifiable.
Each day the children are busy with their hands, and when the light fails to slip between buildings, they sit indoors; their faces blinking blue with stimulation. Every bounded infinite point set has at least one limited point.