Integration with variables Notes_Part_3

# Integration with variables Notes_Part_3 - Figure 7.4 Real...

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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) π 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 = θ increases by 2 π . 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 θ in polar coordinates, we find that √ z = √ r e i θ = √ r e i θ/ 2 = √ r parenleftbigg cos θ 2 + i sin θ 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 θ....
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Integration with variables Notes_Part_3 - Figure 7.4 Real...

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