ช่วยแสดงวิธีทำอย่างละเอียดเป็นภาษาไทยนะคะ ขอบคุณค่ะ
1. Show that
(a) Log (-ei) = 1- (π/2 ) i ; (b) Log ( 1- i) = ½ ln 2 - (π/4 )i
2. Verify that when n = 0 , + 1 , + 2 , ??
(a) log = 1 + 2nπi ; (b) log i = (2n + ½ ) πi
(c) log ( -1 + √3i ) = ln 2 + 2 ( n + 1/3 ) πi
3. Show that
(a) Log ( 1+ i )2 = 2 log ( 1+ i ) ; (b) Log ( -1 + i )2 ≠ 2 log ( -1 + i )
4. Show that
a. log ( i2 ) = 2 log i when log z = lu r + iθ ( r > 0 , π/4 < θ < 9 π/4 )
b. log ( i2 ) ≠ 2 log i when log z = lu r + iθ ( r > 0 , 3π/4 < θ < 11 π/4)
5. Show that
(a) the set of values of log ( i1/2 ) is ( n + ¼ ) πi ( n = 0 , + 1 , + 2 , ?)
and that the same is true of (½
log i
(b) the set of values of log ( i2 ) is not the same as the set of
values of 2 log i
6. Find all roots of the equation log z = iπ/2 (Ans. z = i)
7. Show that
(a) the function Log ( z ? i ) is analytic everywhere except on the half line y = 1( x ≤ 0)
(c) the function
Log ( z + 4 )
z2 + i
is analytic everywhere except at the points + ( 1 ? i )/√2 and on the portion
x ≤ -4 of the real axis.
8. Show that
Re [ log ( z ? 1 ) ] = ½ ln [ (x - 1)2 + y2 ] ( z ≠ 1 )
Why must this function satisfy Laplace?s equation when z ≠ 1 ?
9. Show in two ways that the function ln ( x2 + y2) is harmonic in every domain that does not contain the origin