Integration with variables Notes_Part_3

Integration with variables Notes_Part_3 - Figure 7.4 Real...

Info iconThis preview shows pages 1–2. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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....
View Full Document

This note was uploaded on 02/10/2012 for the course MATH 5587 taught by Professor Olver during the Fall '10 term at University of Central Florida.

Page1 / 3

Integration with variables Notes_Part_3 - Figure 7.4 Real...

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online