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Chapter 7: Problem 7

Find the principal value of \(i^{i}\). Rewrite the base, \(i\), as an exponential first.

Short Answer

Expert verified
\(i^{i}\) is equal to \(e^{-\frac{\pi}{2}}\) or approximately 0.20788 when calculated to 5 decimal places.

Step by step solution

01

Represent \(i\) in polar form

We know that \(i\) is equal to \(e^{\frac{\pi}{2}i}\) when represented in polar form. This is because \(i\) lies on the imaginary axis and the argument (the angle it makes with the positive real axis, measured in the counter-clockwise direction) is \(\frac{\pi}{2}\) (or 90 degrees).
02

Substitute \(i\) with its Exponential Equivalent in the Expression \(i^{i}\)

Replace \(i\) with \(e^{\frac{\pi}{2}i}\) in the equation \(i^{i}\). This gives us \((e^{\frac{\pi}{2}i})^{i}\). Then using the power rule of exponents, we get \(e^{-\frac{\pi}{2}}\).
03

Evaluate the expression

We evaluate \(e^{-\frac{\pi}{2}}\). This is done by calculating the value using the exponential function in a calculator and rounding to some decimal places or by leaving it as is. The principal value is found to be approximately 0.20788, but the principal value in terms of \(e\) is more exact and would be \(e^{-\frac{\pi}{2}}\)

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Most popular questions from this chapter

Write the following in standard form. a. \((4+5 i)(2-3 i)\) b. \((1+i)^{3}\) c. \(\frac{5+3 i}{1-i}\).

Evaluate the following integrals: a. \(\int_{C} \bar{z} d z\), where \(C\) is the parabola \(y=x^{2}\) from \(z=0\) to \(z=1+i\). b. \(\int_{C} f(z) d z\), where \(f(z)=2 z-\bar{z}\) and \(C\) is the path from \(z=0\) to \(z=2+i\) consisting of two line segments from \(z=0\) to \(z=2\) and then \(z=2\) to \(z=2+i\) c. \(\int_{C} \frac{1}{x^{2}+4} d z\) for \(C\) the positively oriented circle, \(|z|=2\). [Hint: Parametrize the circle as \(z=2 e^{i \theta}\), multiply numerator and denominator by \(e^{-i \theta}\), and put in trigonometric form.]

Find series representations for all indicated regions. a. \(f(z)=\frac{z}{z-1},|z|<1,|z|>1\). b. \(f(z)=\frac{1}{(z-1)(2+2)},|z|<1,1<|z|<2,|z|>2\). [Hint Use partial fractions to write this as a sum of two functions first.]

. For the following, determine if the given point is a removable singularity, an essential singularity, or a pole (indicate its order). a. \(\frac{1-\cos z}{z^{2}}, \quad z=0\) b. \(\frac{\sin 2}{z^{2}}, \quad z=0\) c. \(\frac{z^{2}-1}{(z-1)^{2}}, \quad z=1\). d. \(z e^{1 / z}, \quad z=0\). e. \(\cos \frac{\pi}{\pi-\pi}, \quad z=\pi\)

Show that $$ \int_{C} \frac{d z}{(z-1-i)^{n+1}}=\left\\{\begin{array}{cl} 0, & n \neq 0 \\ 2 \pi i, & n=0 \end{array}\right. $$ for \(C\) the boundary of the square \(0 \leq x \leq 2,0 \leq y \leq 2\) taken counterclockwise. [Hint: Use the fact that contours can be deformed into simpler shapes (like a circle) as long as the integrand is analytic in the region between them. After picking a simpler contour, integrate using parametrization.].
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