NPTEL IIT Guwahati

1. Find the values of which make the function $f(z) = \exp(z)$ (a) purely real and (b) purely imaginary.
2. Prove that where $\Im(z)$ denotes the imaginary part of . Deduce that tends to $\infty$ as .
(Note that $\sin(z)$ is an unbounded complex-valued function of a complex variable whereas $\sin(x)$ for $x \in \QTR{Bbb}{R}$ is a bounded real-valued function of a real variable.).
3.Find all solutions of $\exp(z-1) = 1$ .
4. Describe the image of the following sets in the -plane under the mapping $w = \sin(z)$ .
(i)
(ii)
(iii)
(iv)
(v)
(vi)
(vii)

(Note that mappings by $\cos z$ , $\sinh z$ and $\cosh z$ closely related to the $\sin z$ function are easily obtained once mappings by the sine function are known. Because, , and and they are the same as the sine transformation preceded by translation or rotation.

5. Evaluate the following:
(i) $\log(3 - 2i)$ $\qquad$ (ii) $\qquad$ (iii) $(i)^{(-i)}$
6. Determine the domain of analyticity for the function and compute $f^{\prime}(z)$ .
7. Find the principal branch of the function $\log(2z-1)$


Elementary Analytic Functions and their Mapping Properties		Print this page

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1. Find the values of which make the function $f(z) = \exp(z)$ (a) purely real and (b) purely imaginary. 2. Prove that where $\Im(z)$ denotes the imaginary part of . Deduce that tends to $\infty$ as . (Note that $\sin(z)$ is an unbounded complex-valued function of a complex variable whereas $\sin(x)$ for $x \in \QTR{Bbb}{R}$ is a bounded real-valued function of a real variable.). 3.Find all solutions of $\exp(z-1) = 1$ . 4. Describe the image of the following sets in the -plane under the mapping $w = \sin(z)$ . (i) (ii) (iii) (iv) (v) (vi) (vii) (Note that mappings by $\cos z$ , $\sinh z$ and $\cosh z$ closely related to the $\sin z$ function are easily obtained once mappings by the sine function are known. Because, , and and they are the same as the sine transformation preceded by translation or rotation. 5. Evaluate the following: (i) $\log(3 - 2i)$ $\qquad$ (ii) $\qquad$ (iii) $(i)^{(-i)}$ 6. Determine the domain of analyticity for the function and compute $f^{\prime}(z)$ . 7. Find the principal branch of the function $\log(2z-1)$

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