14. Mappings by the exponential function. We shall introduce and develop properties of a number of elementary functions which do not involve polynomials. That chapter will start the exponential function which can also be written is suggested by the familiar additive property
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which can also be written
is suggested by the familiar additive property
of the exponential function in calculus.
can be written , where , according to equation (1). Thus, if
, transformation (2) can be expressed in the form
Let us show that the transformation w=ez maps the rectangular region a≤ x ≤ b, c≤ y ≤ d onto the region ea ≤ρ ≤eb, c ≤ ψ ≤ d. The two regions and corresponding parts of their boundaries are indicated in Fig.21. The vertical line segment AD is mapped onto the arc ρ =ea , c≤ψ≤d, which is labeled A'D' . The images of vertical line segments to the right of AD and joining the horizontal parts of the boundary are larger arcs; eventually, the image of the line segment BC is the arc ρ = eb , c ≤ ψ ≤ d, labeled B'C'. The mapping is one to one if d-c <2π. In particular, if c=0 and d = π , then 0 ≤ ψ ≤ π; and the rectangular region is mapped onto half of a circular ring, as shown in Fig.8, Appendix 2.
when ω =ez , the image of the infinite strip 0 ≤ y ≤ π is the upper half v ≥0 of the ω plane (Fig.22). This is seen by recalling from Example l how a horizontal line y =c is transformed into a ray ψ=c from the origin. As the real number c increases from c =0 to c = π ,the y intercepts of the lines increase from 0 to π and the angles of inclination of the rays increase from ψ =0 to ψ = π . This mapping is also shown in Fig.6 of Appendix 2,where corresponding points on the boundaries of the two regions are indicated.