qr code reader java app Again, setting = sin 1 z we have ei sin in Java

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Again, setting = sin 1 z we have ei sin
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Complex Variables Demysti ed
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= cos(sin 1 z ) + i sin(sin 1 z )
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= 1 z 2 + iz Taking the natural logarithm of both sides, we obtain the desired result: i sin 1 z = ln(iz 1 z 2 ) sin 1 z = i ln(iz 1 z 2 ) EXAMPLE 19 Show that e ln z = rei SOLUTION We use the fact that = + 2n for n = 0,1, 2, to get e ln z = e ln( re
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= e ln r +ln( e = e ln r +i
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= e ln r +i ( +2 n ) = rei ei 2 n = rei (cos 2 n + i sin 2 n ) = rei EXAMPLE 110 Find the fourth roots of 2 SOLUTION We nd the nth roots of a number a by writing r n ein = a ei 0 and equating moduli and arguments, and repeating the process by adding 2 This may not be clear, but we ll show this with the current example First we start out with (rei )4 = 2ei 0 r = 21/ 4
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This is the rst of four roots The second root is (rei )4 = r 4 ei 4 = 2ei 2 r = 21/ 4
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Complex Numbers
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So the second root is z = 21/ 4 ei / 2 = 21/ 4 [cos( / 2) + i sin( / 2)] = i 21/ 4 Next, we have (rei )4 = r 4 ei 4 = 2ei 4 r = 21/ 4 And the root is z = 21/ 4 ei = 21/ 4 (cos + i sin ) = 21/ 4 The fourth and nal root is found using (rei )4 = r 4 ei 4 = 2ei 6 3 = r = 21/ 4 2 In Cartesian form, the root is 3 3 z = 21/ 4 ei = 21/ 4 cos + i sin = i21/ 4 2 2
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Summary
The imaginary unit i = 1 can be used to solve equations like x 2 + 1 = 0 By denoting real and imaginary parts, we can construct complex numbers that we can add, subtract, multiply, and divide Like the reals, the complex numbers form a eld These notions can be abstracted to complex variables, which can be written in Cartesian or polar form
Quiz
1 What is the modulus of z = 2 Write z = 1 i 4 2 4i in standard form z = x + iy 3 + 2i i 5
Complex Variables Demysti ed
3 Find the sum and product of z = 2 + 3i , w = 3 i 4 Write down the complex conjugates of z = 2 + 3i , w = 3 i i 5 Find the principal argument of 2 2i 6 Using De Moivre s formula, what is sin 3 7 Following the procedure outlined in Example 17, nd an expression for sin( x + iy ) 8 Express cos 1 z in terms of the natural logarithm 9 Find all of the cube roots of i 10 If z = 16ei and w = 2ei / 2, what is z w
Functions, Limits, and Continuity
In the last chapter, although we saw a couple of functions with complex argument z, we spent most of our time talking about complex numbers Now we will introduce complex functions and begin to introduce concepts from the study of calculus like limits and continuity Many important points in the rst few chapters will be covered several times, so don t worry if you don t understand everything right away
Complex Functions
When we write z, we are denoting a complex variable, which is a symbol that can take on any value of a complex number This is the same concept you are used to from real variables where we use x or y to represent a variable We de ne a function of a complex variable w = f ( z ) as a rule that assigns to each z a complex
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