Advanced Calculus Midterm 3rd Semeter

Definition of complex number, conjugate number, modulus of complex number,

• A complex number is an ordered pair of real number, denoted as (a,b) or by a + bi, where i² = -1

• If z = x + iy is any complex number, them its conjugate is defined by \bar z = x - yi

• The modulus of a complex number z = x + iy is defined by |z| = \sqrt{x² + y²}

Euler′s formula, exponential form of complex numbers (slide 7,11,14,18, 37

• Any nonzero complex number can be expressed by the form z = r (cos Theta + i * sin Theta), where r = |z| and Theta is an angle between the positive real axis and the vector z.

• Any nonzero complex number can be expressed by the form z = r e^(i * Theta), where r = |z| and Theta is an angle between the real positive axis and the vector z.

Operations with complex numbers (slide 9,10,15-16, 23-24,26,27-28)

Properties f complex number, conjugate number, exponential form, and Euler’s formulas (slide 9, 12, 13-14, 37-39)

• z * conj(z) = |z|²

• Re(z) = (z + conj(z)) / 2; Im(z) = (z - conj(z)) / 2i

• conj(z) = r(cos Theta - i * sin Theta) = r e^(-i Theta)

• conj(z)^n = r * e^(-i Theta n)

On the Complex plane in respect to the points z = x_1 + i*y_1 in C and to the radius r, whether the point inside or outside of the circle can be determine using the inequality (x - x_1)² +(y - y_1)² = r²

For a complex number a = x + iy and positive real number p, the p-neighbourhood of x is N_p(a) = {z in C : |z-a|<p}. This is an open disk with radius p.

A subset S in C is said to be open if for every a in S there’s a real number p in R such that N_p(a) subset of C

Definition of complement of a subset 𝑺 ⊆ ℂ. When is this subset closed? (slide 13)

A subset is a closed set if C \ S is open set.
z in C is a limit point of S if for all epsilot-neighbourhoods of z there exists such point z_0 (inside that neighbourhood) that belongs to S and z ≠ z_0

A subset S in C is closed if and only if S contains all its limit points.

Definition of complex function domain and range (slides 15-16)

For a complex function f:D subset C → C, D is domain, f(z) is range

Definition of complex function (real and imaginary part). (slides 17-18)

Complex function f(z) is equivalent to a pair of real functions u(x,y) and v(x,y) each depending on the two real variables x and y.

Definition of complex function f(z) limit. When the function f(z) it is continuous in a domain? (slides 30-31)

The function is said to be continuous in a domain if it’s continuous at each point of this domain

Write the statement the limit of complex function is unique and proof it. Properties of the limits (slides 35-37)

Properties of limits:

for lim f(z) = l and lim g(z) = g approaching z_0:

• lim( f(z) g(z) ) = l * g

• lim ( f(z) \pm g(z) ) = l \pm g

Definition of the continuous function. (slide 38)

Write and proof the proposition of composition two continuous functions. (slides 40- 43)