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Rajan took over the reins of the central bank from D Subbarao at a time when there is a slump in confidence in both the economy and the currency.
A suave, unflappable former chief economist at the International Monetary Fund (IMF), Rajan enters office as the economy struggles with decade-low growth, a record current account deficit and a steep fiscal shortfall.
The worries about the sluggish economy, and a lack of confidence in how policymakers have addressed it so far, have pummelled the rupee. The currency fell as low as 68.62 on Wednesday, not far from a record low of 68.85 hit a week ago.
However, suspected intervention from the central bank, which was seen selling dollars, helped support the rupee somewhat, currency traders said.
"The rupee will be Rajan's first and key challenge. His IMF aura may help but he will need to win the market's faith by announcing something which helps bring in dollar inflows," said Vikas Babu Chittiprolu, a senior foreign exchange dealer at state-run Andhra Bank.
India's economy is reeling mainly from a dearth of investment and a slowdown in manufacturing activity and consumer demand.
Several banks, including Goldman Sachs this week, have cut their GDP growth forecasts to well below the decade low of 5% for the year ended in March.
Traders say the economy is being further hit by the extraordinary measures from the RBI under Governor Duvvuri Subbarao, which chose to drain cash and raise short-term interest rates in a bid to defend the rupee.
The big question is whether Rajan, who most recently was an advisor at the Finance Ministry, will take the helm of the RBI with a whisper or a bang, and whether he will dismantle any of the mishmash of current central bank measures.
Investors are showing little faith the government can push through substantial reforms, such as a hike in subsidised fuel prices, which could help revive confidence in the economy.
Singh, in a statement ahead of a trip to Russia to attend the Group of 20 nations' summit on Thursday and Friday, said India would also need a more stable global environment.
"The Summit comes at a time when we in India have introduced several reform measures and taken steps to strengthen macro-economic stability, stabilise the rupee and create a more investor friendly environment," Singh said.
"At the same time, a stable and supportive external economic environment is also required to revive economic growth.
Global markets are weakening after leaders of a US Senate panel said they reached an agreement on Tuesday on a draft authorisation for the use of military force in Syria, paving the way for a vote by the committee on Wednesday.
Worryingly for India, global crude and gold prices are surging. Oil and gold are the country's two biggest imports, and are a big factor behind the wide current account deficit that is putting pressure on the rupee.
The prospect that the Federal Reserve will unveil a plan after its policy meeting on Sept 17-18 to start winding down its monetary stimulus is also weighing on emerging markets, but India has fared worse than most because of the lack of confidence it can address its precarious deficits.
