1. Find all the values of log(1-i) , log(3-2i)
2. Find all the values for the following expressions.
(a) 5^i 1 (b) log(1+i)^πi
1-i = √2 e^(-π/4 i)
so, log(1-i) = 1/2 log2 - (π/4 +2kπ)i
Do 3-2i in like wise
5^i = e^(ln5 i) = cos ln5 + i sin ln5
log(1+i)^(πi) = πi log(1+i) = πi (1/2 log2 + π/4 i) = -π^2/4 + π/2 log2 i
Math
To find the values of logarithms involving complex numbers, we can use the properties of complex logarithms. The complex logarithm can be written as log(z) = ln|z| + i(arg(z) + 2πn), where arg(z) is the principal argument of z and n is an integer.
1. Find all the values of log(1-i):
To find the values of log(1-i), we first need to convert 1-i to polar form. Let's call 1-i as z.
z = 1 - i
The magnitude (r) of z is given by |z| = √(1^2 + (-1)^2) = √2
The principal argument (θ) can be found using the formula θ = arctan(-1/1) = -π/4
Now, we can write z in polar form as z = √2 * e^(-iπ/4)
Using the complex logarithm property mentioned earlier, the values of log(1-i) will be:
log(1-i) = ln|√2| + i(arg(√2) + 2πn)
= ln(√2) + i(-π/4 + 2πn)
= ln(2)/2 + i(-π/4 + 2πn), where n is an integer.
2. Find all the values for the following expressions:
(a) 5^i^1:
To calculate this expression, we need to evaluate 5^i. We can use Euler's formula to evaluate it.
Euler's formula: e^(ix) = cos(x) + i*sin(x)
Using Euler's formula, we can write:
5^i = e^(i * ln(5))
Therefore, the expression 5^i^1 becomes:
5^i^1 = (e^(i * ln(5)))^1 = e^(i * ln(5)).
(b) log(1+i)^πi:
To calculate this expression, we need to evaluate log(1+i). Similar to the first part, we can write it in polar form.
1+i = √2 * e^(iπ/4)
Using the complex logarithm property, the expression becomes:
log(1+i)^πi = (ln|√2| + i(arg(√2) + 2πn)) * πi
= (ln(√2) + i(π/4 + 2πn)) * πi
= πi * ln(2)/2 + π^2i^2/4 + 2π^2in, where n is an integer.
Simplifying π^2i^2/4, we get:
π^2i^2/4 = -π^2/4
Therefore, the expression log(1+i)^πi becomes:
log(1+i)^πi = πi * ln(2)/2 - π^2/4 + 2π^2in, where n is an integer.
To find the values of logarithmic expressions, we need to use the complex logarithm and the properties of complex numbers.
1. Logarithm of complex numbers:
The complex logarithm is defined as follows:
log(z) = log|z| + i(arg(z) + 2πk), where k is an integer.
For the given expressions:
(a) log(1-i):
Using the properties of complex logarithm, we can write
log(1-i) = log|1-i| + i(arg(1-i) + 2πk).
To find the value of log|1-i|, we use the modulus of 1-i:
|1-i| = √(1^2 + (-1)^2) = √(1 + 1) = √2.
To find the argument of 1-i, we use the inverse tangent function:
arg(1-i) = arctan(-1/1) = -π/4.
Therefore, log(1-i) = log√2 + i(-π/4 + 2πk), where k is an integer.
(b) log(3-2i):
Similarly, we can calculate:
log(3-2i) = log|3-2i| + i(arg(3-2i) + 2πk).
To find the value of log|3-2i|, we use the modulus of 3-2i:
|3-2i| = √(3^2 + (-2)^2) = √(9 + 4) = √13.
To find the argument of 3-2i, we use the inverse tangent function:
arg(3-2i) = arctan((-2)/3) ≈ -0.588 radians (approximately).
Therefore, log(3-2i) = log√13 + i(-0.588 + 2πk), where k is an integer.
2. Exponential and logarithmic expressions with complex numbers:
(a) 5^i:
To find the value of 5^i, we use Euler's formula:
5^i = e^(i * log(5)).
Using the logarithmic expression log(5) found in part 1, we can calculate:
5^i = e^(i * (log|5| + i(arg(5) + 2πk))).
Simplifying further, we get:
5^i = e^(-arg(5) + 2πik) * e^(i * log|5|).
(b) log(1+i)^πi:
To find the value of log(1+i)^πi, we use the power rule of logarithms:
log(1+i)^πi = πi * log(1+i).
Using the logarithmic expression log(1+i) found in part 1, we can calculate:
log(1+i)^πi = πi * (log|1+i| + i(arg(1+i) + 2πk)).
Simplifying further, we get:
log(1+i)^πi = πi * (log√2 + i(-π/4 + 2πk)).