在帶有電荷黑洞的背景下，又稱帶電黑洞為萊斯納 (Hans Reissner)- 諾斯壯 (Gunner Nordström) 黑洞 (萊-諾黑洞) ，我們考慮俱有質量與電荷量的測試純量場，被吸入萊- 諾黑洞中。藉由量子力 學理論的概念，測式純量場將激發黑洞，使黑洞發射出訊息。利用克萊恩- 戈登 (Klein- Gordon) 方程式的解析和推算，我們將獲得發射出的訊息，其為一條波動方程式。此波動方程乃為二階常微分 波動方程 (ODE)，其形式與匯合式的休恩 (confluent Heun) 微分方程式雷同。但不幸的事，休恩C (HeunC) 微分方程式到目前為止，還沒有精確的解析方程解，因此，我們嘗試找出一組近似波動函數解。假設一組近似波函動數數列，使波動函數數列必須滿足二階常微分波動方程 。利用聲納回傳探測的想法，允許我們考慮無邊界條件下，解析射出的訊息為何，此邊界條件又稱為類駐波邊界 (quasi-normal boundary) 條件。當波動函數滿足波動方程和無邊界條件後，我們將獲得一組三項遞迴關係式。最後，利用類似泰勒展開式的方法，展開和估算遞迴關係式的高階項，並解遞迴關係式的特徵值，則可得到波動方程式的近似波動頻率。

Consider the Klein-Gordon (KG) equation with a massive and charged scalar field in the Reissner- Nordstrom (RN) black hole, we obtain a wave equation which is a second-order ordinary differential equation (ODE). The wave equation is identified with the confluent Heun differential equation which, unfortunately, does not have the exactly solution so far. Hence, here we consider the quasi-normal boundary and present a method which is the three-terms of recurrence relation to approximate the frequency of wave equation.

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