# Question Solved1 Answer by complex 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: 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: 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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