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Browse College Imaginary/Complex Numbers
Stars indicate particularly interesting answers or good places to begin browsing.

Selected answers to common questions:
    DeMoivre's theorem.

Functions of Imaginary Numbers [7/31/1996]
Does (ln i) itself exist? Where does e^iA = cos A + i sin A come from?

Log of Complex Number [9/15/1996]
What does the log of a complex number mean? What are such log laws?

Why do we Bother to Find Roots of Unity? [7/23/1996]
Why do we bother to find the roots of unity? Why not just the simple number? Any significance there?

Asin/acos/atan for Complex Numbers [3/27/1996]
How do you find asin(x+iy), acos(x+iy), and atan(x+iy)?

Calculus of Complex Numbers [10/10/1997]
How do you use the Newton-Raphson method on an equation with complex numbers?

Closed Form of Complex Function [03/24/2003]
I would like a closed form (not a power series) for f(z) such that f is analytic and f(z) = 0 when z = (k*pi)^3, z = ((k*pi)^3)e^i2*pi/3, and z = ((k*pi)^3)e^i4*pi/3 where k is a positive integer.

Complex Analysis Geometry Proof [09/28/2004]
I'm looking for a proof that arg z1 + arg z2 = arg(z1z2).

Complex Analytic Functions [12/08/1998]
I'm trying to find out if abs(z)*(conjugate z) is analytic using the Cauchy-Riemann equations.

Complex Cube Roots of Unity and Simplifying [05/17/2005]
With w denoting either of the two complex cube roots of unity, find [(2w + 1)/(5 + 3w + w^2)] + [(2w^2 + 1)/(5 + w + 3w^2)], giving your answer as a fraction a/b, where a, b are integers with no factor in common.

Complex Equations [6/14/1996]
Let z be an element of the complex numbers...

Complex Integrals and the Residue Theorem [12/18/2000]
How can I calculate the integral over C of (z^2/((z-1)^2*(z+1)))dz, where C is the circle C = {z| |z - 2i| = 2}? Can I use the Taylor Series?

Complex Numbers [03/11/2003]
z^4 + z^3 + z^2 + z + 1 = 0

Complex Numbers and Supremum Property [12/06/2002]
Is it possible to define a total order on C for which one does have the supremum property?

Complex Numbers to Complex Powers [10/19/2000]
Can Euler's equation be used to find any number raised to a complex power? How is it possible that all real numbers raised to an imaginary power map to the complex unit circle?

Complex Powers [04/10/2002]
Given e^(2*pi*i/2*pi*i) = e^(1) = e ... 1^(1/2*pi*i) has to be equal to e. I am having trouble proving this last step.

Complex Variables [03/25/2003]
Is there any complex root for an equation like sin(x)=3/2? What does a^i= ? where a is a real constant.

Conjectures vs. Hypotheses [01/12/1999]
What is the difference between the terms 'conjecture' and 'hypothesis'? Should the Riemann hypothesis be the Riemann conjecture?

Conjugate Roots of Complex Numbers [12/01/2000]
If you take the nth root of a complex number, is there a way to tell if there will be any conjugate roots among the n answers?

Convergence of Product of Sines [10/17/2003]
Prove that (sin(pi/n))*(sin(2pi/n))*...*(sin((n-1)pi/n)) = n/(2^(n-1)) for n >= 2.

Cube Roots of Numbers [11/05/1997]
If you take i (sqrt(-1)), the cube root is -i, but since x^3 = i is degree three there should be three different values of x. What are they?

DeMoivre's Theorem: Standard Form [3/19/1996]
Use DeMoivre's theorem to write (1-i)^10 in standard form.

Deriving Lagrange's Trig Identity [01/09/2004]
Using the identity 1 + z + z^2 + ... + z^n = (1 - z^(n+1))/(1 - z), z not = 1, derive Lagrange's trig identity: 1 + cosx + cos(2x) + ... + cos(nx) = 1/2 + (sin[(2n+1)x/2])/(2sin(x/2)) where 0 < x < 2*pi.

e^(pi*i) = -1: A Contradiction? [8/17/1996]
I know that e^(i*Pi) = -1. But squaring and taking a natural log of both sides, you get 2*i*Pi = 0. Please explain.

Euler in the Product of a Regular Polygon's Diagonal Lengths [04/06/2010]
A professor emeritus considers an n-sided regular polygon A1, A2, ... An inscribed in the unit circle; and conjectures that the product of the lengths of its diagonals equals n. By defining the polynomial f(x) as the product of x - r over its (n - 1) roots, and applying complex numbers and Euler's equation, Doctor Vogler proves that sin(pi/n) * sin(2pi/n) * ... * sin[(n - 1)pi/n] = n/2^(n - 1).

Euler's Equation: First Step [05/18/1999]
Can any complex number can be expressed as cos(t)+i*sin(t)?

Exponentiation [08/16/1997]
How do I calculate x^y using only exp, ln, log, and the trigonometric functions?

Factoring Polynomials over Real and Complex Numbers [07/17/2006]
I am having difficulties factoring polynomials like x^4 - 15x^2 - 75. It is irreducible over the integers but its graph suggests there are in fact roots. How can I factor over the real and complex numbers?

Find Complex Numbers [12/16/1995]
Find all complex numbers such that (conjugate z)(z)^(n-1) = 1.

Finding Arctan of a Complex Number [08/18/2008]
Can we separate the real and imaginary parts of arctan(x + iy)?

Finding GCD of Complex Numbers with Euclidean Algorithm [10/11/2004]
I would like to calculate GCD(135 - 14i, 155 + 34i) via the Euclidean algorithm, but I don't know how to do that with complex numbers.

Finding Roots of Complex Numbers [09/01/2005]
How do you find the nth roots of a complex number a + bi?

Finding Roots of Polynomials with Complex Numbers [09/27/2001]
I read in the archives that you can find the roots of 3rd or higher- degree polynomials with complex numbers...

Find the Flaw [08/02/2001]
I don't understand where the following proof goes wrong...

Getting Complex with Euclid, Sturm, and Newton [02/20/2016]
An adult seeks a method for determining a polynomial's complex roots. Doctor Vogler offers some clarifications, iterative techniques — and caveats.

Graphing Complex and Real Numbers [02/26/2003]
Since on the Cartesian plane we can only graph real zeros and real solutions, are we truly graphing the function when we omit the complex and imaginary zeros and solutions?

Graphing Complex Functions [08/11/1998]
In the quadratic equation y = x^2 + 5x + 12, when y = 0 has no solutions, where (if anywhere) do these numbers lie on the graph of this equation?

Imaginary Numbers Raised to Imaginary Numbers [12/29/2001]
I input i^i into my TI-89 graphing calculator, and the calculator returned e^(-pi/2). Why?

Inverse of arg(z) [10/10/2003]
What is the inverse of the function arg(z)?

Is There a Universal Set of All Numbers? [06/16/2004]
The real numbers and the imaginary numbers are subsets of the complex numbers. Is the set of complex numbers a subset of a more universal set? Is there a universal set of all numbers agreed upon today?

The ith Root of -1 [09/16/1999]
Why does the ith root of -1 equal 23.14069...?

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