Question Solved1 Answer   2. Use the theorem in Sec. 23 to show that \( f^{\prime}(z) \) and its derivative \( f^{\prime \prime}(z) \) exist everywhere, and find \( f^{\prime \prime}(z) \) when (a) \( f(z)=i z+2 \) (b) \( f(z)=e^{-x} e^{-i y} \); (c) \( f(z)=z^{3} \); (d) \( f(z)=\cos x \cosh y-i \sin x \sinh y \). Ans. (b) \( f^{\prime \prime}(z)=f(z) \); (d) \( f^{\prime \prime}(z)=-f(z) \)

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Transcribed Image Text: 2. Use the theorem in Sec. 23 to show that \( f^{\prime}(z) \) and its derivative \( f^{\prime \prime}(z) \) exist everywhere, and find \( f^{\prime \prime}(z) \) when (a) \( f(z)=i z+2 \) (b) \( f(z)=e^{-x} e^{-i y} \); (c) \( f(z)=z^{3} \); (d) \( f(z)=\cos x \cosh y-i \sin x \sinh y \). Ans. (b) \( f^{\prime \prime}(z)=f(z) \); (d) \( f^{\prime \prime}(z)=-f(z) \)
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Transcribed Image Text: 2. Use the theorem in Sec. 23 to show that \( f^{\prime}(z) \) and its derivative \( f^{\prime \prime}(z) \) exist everywhere, and find \( f^{\prime \prime}(z) \) when (a) \( f(z)=i z+2 \) (b) \( f(z)=e^{-x} e^{-i y} \); (c) \( f(z)=z^{3} \); (d) \( f(z)=\cos x \cosh y-i \sin x \sinh y \). Ans. (b) \( f^{\prime \prime}(z)=f(z) \); (d) \( f^{\prime \prime}(z)=-f(z) \)
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