Let a function f be analytic everywhere in a domain D. Suppose that f(z) is pure imaginary for all z in D. What can we conclude about the values of f(z)? 1. (Hint: Use the theorem or the first corollary presented in Lecture 10)
Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
![1.
Let a function f be analytic everywhere in a domain D. Suppose that f(z) is pure
imaginary for all z in D. What can we conclude about the values of f(z)?
(Hint: Use the theorem or the first corollary presented in Lecture 10)
Let f and g be analytic functions in a domain D. If f'(z) = g'(z) for all z in D,
then show that f(z)= g(z)+c, where c is a complex constant.
2.
3.
Let u(x, y) = x² –- y² and v(x, y) = x' - 3xy. Show that u and v are harmonic
functions but that their product uv is not harmonic.
Show that u(x,y) = 2x-x' +3xy is harmonic and find a harmonic conjugate
v(x, y).
4.
Show that exp(z) s exp(z|") for all z e C.
5.
6.
Show that Log[(-1+i)*]# 2Log(-1+i).
7.
Find all roots of the equation log(z) = ri/2
8.
Find the principal value of (1+ i)'.
9.
Use the definitions of sin(z) and cos(z) given in Lecture 13 at the 13:30 mark to
= cos(z) for all z E C.
prove that Sin] z+
(Do not use any other trigonometry identities for question 9)
10.
Evaluate (3t-i)² dt](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcfc5bdab-6654-444e-b9d5-5fc4b8470fbb%2F1419c493-5955-49c4-9c66-686c72cbe641%2Fkd7f7dk_processed.jpeg&w=3840&q=75)
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