Solved: TEAM PROJECT. Further Properties of the

Chapter 13, Problem 13.1.131

(choose chapter or problem)

Further Properties of the Exponential Function.

(a) Analyticity. Show that \(e^{z}\) is entire. What about \(e^{1 / z}? e^{\bar{z}} ? e^{x}(\cos k y+i \sin k y)\)? (Use the Cauchy-Riemann equations.)

(b) Special values. Find all z such that (i) \(e^{z}\) is real, (ii) \(\left|e^{-z}\right|<1\), (iii) \(e^{\bar{z}}=\overline{e^{z}}\).

(c) Harmonic function. Show that \(u=e^{x y} \cos \left(x^{2} / 2-y^{2} / 2\right)\) is harmonic and find a conjugate.

(d) Uniqueness. It is interesting that \(f(z)=e^{z}\) is uniquely determined by the two properties \(f(x+i 0)=e^{x}\) and f ‘(z) = f(z), where f is assumed to be entire. Prove this using the Cauchy-Riemann equations.

Text Transcription:

e^z

e^{1/z}? e^{bar{z}}? e^x (cos ky + i sin ky)

|e^-z| < 1

e^{bar{z}} = overline{e^z}

u = e^{xy} cos (x^2/2 -y^2/2)

f(z) = e^z

f(x + i0) = e^x