Final answer:
The principal value of ln(1+i) is ln(√2) + i(π/4). This can be plotted on the complex plane at the point (0.3466, 0.7854).
Step-by-step explanation:
The question asks for the evaluation and plot on the complex plane of the principal value of ln(1+i). To find the principal value of the natural logarithm of a complex number, we use the polar form of the complex number, which involves converting the complex number from its rectangular form a + bi to a polar form r(cosθ + isinθ), where r is the modulus and θ is the argument of the complex number.
For the complex number 1+i, we calculate the modulus r as √(1^2 + 1^2) = √2 and the argument θ as arctan(b/a) = arctan(1/1) = π/4. Therefore, in polar form, 1+i can be written as √2(cos(π/4) + isin(π/4)). The principal value of ln(1+i) is then given by ln r + iθ, which is ln(√2) + i(π/4).
To plot ln(1+i) on the complex plane, we identify the real part ln(√2) and the imaginary part π/4. The real part corresponds to the horizontal axis and the imaginary part corresponds to the vertical axis. We would plot this point at approximately (0.3466, 0.7854) on the complex plane, where 0.3466 is the decimal approximation of ln(√2) and 0.7854 is the decimal approximation of π/4.