Question

suppose $\large\rm { [a,b] \subseteq \mathbb{R}}$ , and let $\large\rm { \varphi}$ be a continuous complex valued function on the product space $\large\rm { \Omega \times [a,b]}$ assume that for each $\large\rm { t \in [a,b]}$ , the function $\large\rm { z \longrightarrow \varphi (z,t)}$ is analytic on $\large\rm{ \Omega}$ define F on Ω by $\large\rm { F(z) = \displaystyle\int_{a}^{b} \varphi (z,t) dt, z \in \Omega}$ and F is analytic on Ω and $\large\rm { F'(z) = \displaystyle\int_{a}^{b} \frac{\partial \varphi}{\partial z} (z,t) dt , z \in \Omega}$​

Answer 1

Correct Question:

suppose $\large\rm { [a,b] \subseteq \mathbb{R}}$ , and let $\large\rm { \varphi}$ be a continuous complex valued function on the product space $\large\rm { Ω \times [a,b]}$ assume that for each $\large\rm { t \in [a,b]}$ , the function $\large\rm { z \longrightarrow \varphi (z,t)}$ is analytic on $\large\rm{ Ω}$ define F on Ω by $\large\rm { F(z) = \displaystyle\int_{a}^{b} \varphi (z,t) dt, z \in Ω}$ and F is analytic on Ω and $\large\rm { F'(z) = \displaystyle\int_{a}^{b} \frac{\partial \varphi}{\partial z} (z,t) dt , z \in Ω}$

Proof:

Fix any disk $\large\rm{ D(z_{0} , r)}$ such that $\large\rm { \bar{D} (z_{0}, r) \subseteq Ω}$ . Then for each $\large\rm { z \in D (z_{0} , r)}$,

$\large\rm { F(z) = \displaystyle\int_{a}^{b} \varphi (z,t) dt}$

$\small\rm{ = \frac{1}{2 \pi i} \displaystyle\int_{a}^{b} \Bigg\lgroup \displaystyle\int_{C(z_{0} ,r)} \frac{\varphi (w,t)}{w-z} dw \Bigg\rgroup dt}$

$\small\rm{ = \frac{1}{2 \pi i} \displaystyle\int_{C(z_{0} , r)} \Bigg\lgroup \displaystyle\int_{a}^{b} \varphi(w,t) dt \bigg\rgroup \frac{1}{w-z} dw}$

$\small\rm { F'(z) = \frac{1}{2 \pi i} \displaystyle\int_{C(z_{0} , r)} \Bigg [ \displaystyle\int_{a}^{b} \varphi(w,t) dt \Bigg ] \frac{1}{(w-z)^{2}} dw}$

$\small\rm { = \displaystyle\int_{a}^{b} \Bigg [ \frac{1}{2 \pi i} \displaystyle\int_{C(z_{0} ,r)} \frac{\varphi (w,t)}{(w-z)^{2}} dw \Bigg ] dt}$

$\large\rm { = \displaystyle\int_{a}^{b} \frac{\partial \varphi}{\partial z} (z,t) dt }$

Hence proven.

Answer 2

Answer:

suppose $\large\rm { [a,b] \subseteq \mathbb{R}}$ , and let $\large\rm { \varphi}$ be a continuous complex valued function on the product space $\large\rm { \Omega \times [a,b]}$ assume that for each $\large\rm { t \in [a,b]}$ , the function $\large\rm { z \longrightarrow \varphi (z,t)}$ is analytic on $\large\rm{ \Omega}$ define F on Ω by $\large\rm { F(z) = \displaystyle\int_{a}^{b} \varphi (z,t) dt, z \in \Omega}$ and F is analytic on Ω and $\large\rm { F'(z) = \displaystyle\int_{a}^{b} \frac{\partial \varphi}{\partial z} (z,t) dt , z \in \Omega}$

Step-by-step explanation:

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