I have two problems withcomplexnumbers:
1) Use properties of moduli to sow that when |z3| does not equal |z4|,
Re(z1 + z2) / |z3 + z4| <= (is smaller or equals) (|z1| + |z2|) / (| |z3| -|z4| |)
2) Verify that sqrt(2) * |z| >= |Re z| + |Im z|
(suggestion: reduce this inequality to (|x| - |y|)^2 +. 0
Design class Complex for working withcomplexnumbers of the form a + bi, where i is the square root of -1. Your class must have two overloaded operators for adding and subtracting the complexnumbers. The sum and the difference of two complexnumbers a + bi and c + di is defined as (a+c) + (b+d)i (respectively, (a-c) + (b-d)i).
1. Find the area bounded by the lines y=0, y=2 an y=sqrt(x)
2. Find the partial decomposition of:
3. Find the critical points and the inflection points of the following function:
f(x) = x^4 - 4x^3 + 10
4. Simplify the following complex expressions, expressing each in the form (a + jb)
1.Find an example of a sequence an of complexnumbers such that the series
SUM a_n converges (conditionally), yet the series SUM a^(3)_n diverges.
2.Determine the set of complexnumbers z for which the series SUM(1â?'z^2)^n converges.
SUM means sigma.
Complete only 3.6. See attached file for full problem description.
Let z denote a complex variable
z = x + jy = re^jt
The complex conjugate of z is dentoed by z* and is given by
z* = x - jy = re^-jt
Derive each of the following relations, where z, z1 and z2 are arbitrary complexnumbers:
(a) zz* = r^2
(b) z/z* = e
Prove algebraically that for complexnumbers,
|z1| - |z2| (is less than or equal to) |z1 + z2| (is less than or equal to) |z1| + |z2|
Interpret this result in terms of two-dimensional vectors. Prove that
|z-1| < |sqrt(z^2 - 1) < |z + 1|, for H(z) > 0.
Show that complexnumbers have square roots and that the square roo