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Unformatted text preview: Figure 7.4. Real and Imaginary Parts of z . also have complex logarithms! On the other hand, if z = x < 0 is real and negative, then log z = log | x | + (2 k + 1) i is complex no matter which value of ph z is chosen. (This explains why one avoids defining the logarithm of a negative number in first year calculus!) Furthermore, as the point z circles once around the origin in a counter-clockwise direction, Im log z = ph z = increases by 2 . Thus, the graph of ph z can be likened to a parking ramp with infinitely many levels, spiraling ever upwards as one circumambulates the origin; Figure 7.3 attempts to sketch it. At the origin, the complex logarithm exhibits a type of singularity known as a logarithmic branch point , the branches referring to the infinite number of possible values that can be assigned to log z at any nonzero point. ( f ) Roots and Fractional Powers : A similar branching phenomenon occurs with the fractional powers and roots of complex numbers. The simplest case is the square root function z . Every nonzero complex number z negationslash = 0 has two different possible square roots: z and z . Writing z = r e i in polar coordinates, we find that z = r e i = r e i / 2 = r parenleftbigg cos 2 + i sin 2 parenrightbigg , (7 . 16) i.e., we take the square root of the modulus and halve the phase: vextendsingle vextendsingle z vextendsingle vextendsingle = radicalbig | z | = r , ph z = 1 2 ph z = 1 2 ....
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